su(2)-irreducibly covariant quantum channels and some

SU(2)-Irreducibly Covariant Quantum Channels and Some

Applications

Muneerah Al Nuwairan

Thesis submitted to the Faculty of Graduate and Postdoctoral Studies

in partial fulfillment of the requirements for the degree of Doctor of Philosophy in

Mathematics 1

Department of Mathematics and Statistics

Faculty of Science

University of Ottawa

c© Muneerah Al Nuwairan, Ottawa, Canada, 2015

1The Ph.D. program is a joint program with Carleton University, administered by the Ottawa-Carleton Institute of Mathematics and Statistics

Abstract

In this thesis, we introduce EPOSIC channels, a class of SU(2) -covariant quantum

channels. For each of them, we give a Stinespring representation, a Kraus represen-

tation, its Choi matrix, a complementary channel, and its dual map. We show that

these channels are the extreme points of all SU(2) -irreducibly covariant channels. As

an application of these channels to the theory of quantum information, we study the

minimal output entropy of EPOSIC channels, and show that a large class of these

channels is a potential example of violating the well-known problem, the additivity

problem. We determine the cases where their minimal output entropy is not zero, and

obtain some partial results on the fulfillment of their entanglement breaking property.

We find a bound of the minimal output entropy of the tensor product of two SU(2)

-irreducibly covariant channels. We also get an example of a positive map that is not

completely positive.

ii

Resume

Dans cette these, nous introduisons une classe de canaux quantiques, les canaux

EPOSIC. Pour chacun d’entre eux, nous donnons leur representation de Stinespring,

leur decomposition de Kraus, leur matrice de Choi, leur canal complementaire, et

l’application duale. Nous montrons que ces canaux sont les points extremaux de tous

les canaux irreductibles SU(2)-covariants. En guise d’application de ces canaux a la

theorie de l’information quantique, nous etudions l’entropie minimale de sortie des

canaux EPOSIC, et montrons que beaucoup de ces canaux constituent des exemples

potentiels de violation du celebre probleme d’additivite de l’entropie minimale de

sortie. Nous determinons les canaux pour lesquels l’entropie minimale de sortie est

non nulle, et nous obtenons des resultats partiels pour la propriete de ’entanglement

breaking’ (cassage d’intrication). Nous trouvons une borne sur l’entropie minimale

de sortie du produit tensoriel de deux canaux irreductibles SU(2)-covariants. Nous

obtenons aussi un nouvel exemple d’application positive qui n’est pas completement

positive.

iii

Acknowledgements

This thesis was done under the supervision of professors B. Collins and T. Giordano.

I greatly appreciate their patient advice and help. I would also like to thank the

Saudi Arabia government, and King Faisal University whose support enabled me to

finish my studies and write this thesis.

iv

Dedication

to my husband and children...

v

Contents

Introduction 1

Contributions 4

I The Extreme Points of SU(2)-Irreducibly Covariant

Quantum Channels 6

1 Preliminaries in Representation Theory 7

1.1 Representations of compact groups . . . . . . . . . . . . . . . . . 7

1.2 G-equivariant maps . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.2.1 Basic definitions and results . . . . . . . . . . . . . . . . . . 12

1.2.2 Examples of G-equivariant maps . . . . . . . . . . . . . . . . 19

2 Representations of SU(2) 27

2.1 The irreducible representations of SU(2) . . . . . . . . . . . . . . 28

2.2 Clebsch-Gordan expansion . . . . . . . . . . . . . . . . . . . . . 31

2.3 An SU(2)-equivariant isometry on the representation space of

SU(2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

2.3.1 The isometry αm,n,h . . . . . . . . . . . . . . . . . . . . . . 34

2.3.2 The matrix coefficients of αm,n,h . . . . . . . . . . . . . . . . 37

vi

CONTENTS vii

2.4 An application: The algebra End(Pm) as a direct sum of orthog-

onal SU(2)-subspaces . . . . . . . . . . . . . . . . . . . . . . . . 42

3 Introduction to Quantum Channels 54

3.1 Positive and completely positive maps . . . . . . . . . . . . . . . 54

3.2 Quantum channels . . . . . . . . . . . . . . . . . . . . . . . . . . 56

3.2.1 Quantum systems and quantum states . . . . . . . . . . . . 56

3.2.2 Quantum channels, definition and examples . . . . . . . . . 58

3.2.3 Characterization of quantum channels . . . . . . . . . . . . 61

3.3 G-covariant quantum channels . . . . . . . . . . . . . . . . . . . 64

4 EPOSIC Channels 70

4.1 EPOSIC channels . . . . . . . . . . . . . . . . . . . . . . . . . . 71

4.2 Kraus representations of EPOSIC channels . . . . . . . . . . . . 72

4.3 The Choi matrix of the EPOSIC channel . . . . . . . . . . . . . 78

4.4 A channel complementary to the EPOSIC channel . . . . . . . . 82

4.5 Duals of EPOSIC channels . . . . . . . . . . . . . . . . . . . . . 85

4.6 A positive map that is not completely positive . . . . . . . . . . 88

5 SU(2)-Irreducibly Covariant Channels 92

5.1 Extreme points of SU(2)-irreducibly covariant channels . . . . . 93

5.2 SU(2)-irreducibly covariant channel as direct sum of operators . 101

II The Minimal Output Entropy and The Entanglement

Breaking Property of EPOSIC Channels 112

6 The Minimal Output Entropy and The Entanglement Breaking

Property of Quantum Channels 113

6.1 The minimal output entropy of quantum channels . . . . . . . . 113

CONTENTS viii

6.2 The entanglement breaking property of quantum channels . . . . 117

6.3 The additivity of the classical capacity of quantum channels . . . 120

7 The Minimal Output Entropy and The Entanglement Breaking

Property of EPOSIC Channels 123

7.1 The minimal output entropy of EPOSIC channel . . . . . . . . . 124

7.2 Examples and special cases . . . . . . . . . . . . . . . . . . . . . 127

7.2.1 The minimal output entropy of Φm,1,1 . . . . . . . . . . . . . 128

7.2.2 Lower bound of the minimal output entropy of an element

in QC(P1, Pm)SU(2) . . . . . . . . . . . . . . . . . . . . . . . 130

7.3 The entanglement breaking property of EPOSIC channels . . . . 134

8 The Minimal Output Entropy of the Tensor Product of SU(2)-

Irreducibly Covariant Channels 136

8.1 The tensor product of SU(2)-irreducibly covariant channels. . . . 137

8.2 Bound for the minimal output entropy of the tensor product of

two SU(2)-irreducibly covariant channels. . . . . . . . . . . . . . 139

III Appendices 142

A Background Results in Operator Algebras 143

A.1 Background definitions and lemmas . . . . . . . . . . . . . . . . 143

A.2 Positive and completely positive maps . . . . . . . . . . . . . . . 150

A.3 Covariant Stinespring dilation theorem . . . . . . . . . . . . . . . 153

B Deferred Proofs 157

B.1 Deferred proofs in chapter 2 . . . . . . . . . . . . . . . . . . . . . 157

B.2 Proofs in chapter 5 . . . . . . . . . . . . . . . . . . . . . . . . . . 166

B.3 Proofs in Chapter 7 . . . . . . . . . . . . . . . . . . . . . . . . . 167

CONTENTS ix

C List of Equations That Are Used in The Computation 170

Bibliography 173

Introduction

According to Moore’s law, the power of computers can be doubled for the same cost

each two years [27, p.4]. Scientists believe that Moore’s law might not apply by

the 2020s, due to size difficulties. As a result, a lot of research efforts have been

directed toward computing at the atomic level, where the classical laws don’t apply,

and the need for quantum laws appears. That was the birth of quantum information

theory, which generalizes the classical one. Here is Holevo’s description of quantum

information theory in his address to the ICM in 2006 [17],

The problem of data transmission and storage by quantum informa-

tion carriers received increasing attention during past decade, owing to the

burst of activity in the field of quantum information and computation. At

present we are witnessing emergence of theoretical and experimental foun-

dations of the quantum information science. It represents a new exciting

research field addressing a number of fundamental issues in both quantum

physics and in information and computer sciences. On the other hand, it

provides a rich source of well-motivated mathematical problems.

In analogy to classical information theory, the quantum information one studies the

protocols of quantum transmissions for quantum information. A quantum channel (a

channel) is any method used to transfer the information from one or more quantum

systems to other quantum systems. This transmission of information is not always

accurate; the channel itself as well as the environment create noise which can cause

1

Introduction 2

information loss, and limit the efficiency of the quantum channel. The classical ca-

pacity of the channel is the maximum number of bits that can reliably be sent using

the channel [35]. A fundamental problem in quantum information theory is to de-

termine the classical capacity of a quantum channel. The additivity conjecture in

quantum information is that the classical capacity of a quantum channel is additive,

i.e, running two channels in parallel will not increase their total classical capacity.

A fundamental result of quantum information theory, The quantum coding theorem

[18, 32], shows that the additivity of the classical capacity can be inferred from the

additivity of another quantity, known as the Holevo bound or the Holevo capacity. In

2000, C. King and M. Ruskai [25] introduced the notion of minimal output entropy,

and P. Shor [36] (2004) showed that several conjectures in quantum information the-

ory are all equivalent. In particular, the Holevo capacity is additive if and only if the

minimal output entropy is. In 2008, Hastings [11] was able to show the existence of

a counter-example to the additivity of the minimal output entropy, using a random

construction. However, no explicit example was given.

In this thesis, we present an example of a new class of quantum channels, study their

properties, and their minimal output entropy. The thesis consists of two parts. In

Part I, we introduce the new channels, and study their properties. In Chapter 1, we

review the basic definitions and state all the related propositions and lemmas from

representation theory. In Chapter 2, we review the irreducible representations of

the group SU(2), and define an SU(2)-equivariant isometry. Chapter 3 contains all

needed background results from quantum information theory. Chapter 4 is devoted to

introducing and characterizing the new class of quantum channels, EPOSIC channels

(the Extreme Points Of SU(2)-Irreducibly Covariant channels), to compute Kraus

operators for them, and to find their Choi matrices. The chapter ends by giving an

example of a positive map that is not completely positive, using EPOSIC channels. In

Chapter 5, we study the SU(2)-irreducibly covariant channels, and show that EPOSIC

channels are the extreme points of this set. Part II consist of three chapters, chapter 6

Introduction 3

contains the definitions and all needed results about the minimal output entropy and

the entanglement breaking property for a quantum channel (E.B.T). It also explains

their relation to the additivity of classical capacity of quantum channel. In chapter 7,

we study the minimal output entropy, and E.B.T property of the EPOSIC channels.

Chapter 8 studies the minimal output entropy of the tensor product of two SU(2)-

irreducibly covariant channels.

Contributions

In this section, we list the contributions of the thesis. We hope that our results will

be a significant addition to the field of operator algebra and quantum information

theory. Our main contributions can be summarized as follows:

1. Constructing EPOSIC channels (Proposition 4.1.1), a new class of quantum

channels that form the extreme points of all SU(2)-irreducibly covariant chan-

nels (Corollary 5.1.5).

2. Giving a full description of the EPOSIC channel, by obtaining a Stinespring

representation, a Kraus representation, the Choi matrix of the EPOSIC chan-

nel (Definition 4.2.1, and Proposition 4.3.5), computing a channel complemen-

tary to EPOSIC channel, and computing its dual map (Proposition 4.4.4, and

Proposition 4.5.6).

3. Showing that any SU(2)-irreducibly covariant channel is an orthogonal direct

sum of operators (Corollary 5.2.4).

4. For an SU(2)-irreducible subspace H, we give explicit formulae for the pro-

jections of End(H) into its SU(2)-irreducible invariant subspaces (Proposition

2.4.2).

5. Proving that any completely positive SU(2)-irreducibly equivariant map is a

multiple of an SU(2)-covariant channel (Corollary 5.1.6).

4

Contributions 5

6. Obtaining an example of a positive, non-completely positive map (Proposition

4.6.3, and Proposition 4.6.5).

7. As applications in quantum information theory, we were able to

(a) Determine the EPOSIC channels with zero minimal output entropy (Propo-

sition 7.1.1, and Corollary 7.1.7).

(b) Find a lower bound of the minimal output entropy of some of SU(2)-

irreducibly covariant channels (Proposition 7.2.13).

(c) Examine the entanglement breaking property of EPOSIC channel (Section

7.3).

(d) Obtain an upper bound on the minimal output entropy for the tensor

product of two SU(2)-irreducibly covariant channels (Corollary 8.2.4).

Part I

The Extreme Points of

SU(2)-Irreducibly Covariant

Quantum Channels

6

Chapter 1

Preliminaries in Representation

Theory

The construction and study of the channels we present in this thesis depend heavily

on the representations of the group SU(2). The present chapter contains background

definitions and results from representation theory that are needed for the thesis. For

more details, we refer the reader to [3], [9], [13], [29], [33] and [38]. The definition of

Hilbert spaces, and all related basic mathematical results are in Appendix A. In this

thesis, we assume all vector spaces to be complex vector spaces of finite dimension.

1.1 Representations of compact groups

Definition 1.1.1. A topological group is a set G which has both the structure of a

group and a topological space, such that the group operations (x, y) 7→ xy and x 7→ x−1

are continuous. A compact group is a topological group whose topology is compact.

Examples 1.1.2.

1. Any finite group endowed with the discrete topology is a compact group.

7

1. Preliminaries in Representation Theory 8

2. The unit circle under complex multiplication, and with the usual topology is an

infinite compact group.

3. The real numbers R under the usual addition +, and with the usual topology

is a non-compact topological group.

4. Let H be a finite dimensional complex vector space. The set of all invertible

linear maps A : H −→ H forms a group under composition, called the general

linear group, denoted by GL(H).

5. For n ∈ N, let Mn(C) denote the set of n× n-complex matrices. The subsets

U(n) = T ∈Mn(C) : TT ∗ = In and SL(n) = T ∈Mn(C) : det(T ) = 1,

form subgroups of GL(n,C), called the unitary group and the special linear

group, respectively. The unitary group U(n), and its subgroup, the special

unitary group SU(n) = U(n) ∩ SL(n), are compact topological groups.

Following [30, p.13, 128] and [37, p.21], we have:

Definition 1.1.3. Let G be a group, and H be a vector space.

1. A representation of G in H is a group homomorphism πH : G −→ GL(H).

If H is a Hilbert space, and πH(g) is a unitary operator for each g ∈ G, we say

πH is a unitary representation.

2. If G is a topological group, and H is a Banach space, then a continuous repre-

sentation of G is a representation πH such that the map

G×H −→ H

(g, h) 7−→ πH(g)(h), h ∈ H

is continuous.


The space H is called the representation space of πH. If H is finite dimensional,

we say the representation is finite dimensional. The dimension of H is called the

degree of the representation [37, p.21].

Notation 1.1.4. We denote a representation πH of a group G in a Hilbert space H

by (H, πH)G. If the group G is clear from context then we may omit the subscript G.

Examples 1.1.5.

1. Let G be a group, and H be a vector space. The trivial representation of G

is the map πH : G −→ End(H) defined by taking any element g ∈ G to the

identity map on H.

2. For a locally compact group G there exist a left invariant (Haar) measure µ on

G, where L2(G, dµ) = f : G −→ C :∫|f |2 dµ < ∞ is a Hilbert space. The

left regular representation of G in L2(G, dµ) [30, p.132], is a unitary faithful

representation of G.

3. Let n ∈ N. By representing the vectors in Cn as n × 1 matrices, the groups

U(n) and SU(n) have representations in Cn given by matrix multiplication.

These representations are called the standard representations of U(n) and SU(n)

respectively [3, p.69].

Example 1.1.6. Let (H, πH) be a representation of a group G. The following are

representations of G in H, where H denotes the conjugate space of H (see Appendix

A for definition of H).

1. πH : G −→ GL(H) by πH(g) = πtH(g−1), for g ∈ G.

2. πH : G −→ GL(H) by πH(g) = πH(g), for g ∈ G.

The representations πH and πH are called the contragredient, and the conjugate

representation of πH respectively. They coincide when πH is unitary.


Definition 1.1.7. Let (H, πH) be a representation of a group G. Then

1. A subspace W of H is called G -invariant if πH(g)(W ) ⊆ W for each g ∈ G.

The restriction of πH to a G-invariant subspace is called a subrepresentation of

πH.

2. The space H is G-irreducible if it has no proper nonzero G-invariant subspaces.

In this case, the representation πH is called an irreducible representation.

We restrict the use of the symbol ρ to irreducible representations.

Examples 1.1.8.

1. Any irreducible representation of an abelian group is one dimensional [8, p.71].

2. For n ∈ N, let Sn be the symmetric group of 1, 2, ..., n. The irreducible

representations of Sn are in one-to-one correspondence with the partitions of n

[9, p.44-54].

3. For m,n ∈ N, let P(m,n) be the space of homogenous polynomials of degree m

in n variables x = (x1, x2, ..., xn) over C. The group GL(n,C) has a represen-

tation in P(m,n) given by

π(g)(f(x)) = f(xg)

for f ∈ P(m,n) and g ∈ GL(n,C), where xg denotes matrix multiplication.

For Hilbert spaces H and K, let End(H,K) denote the vector space of linear

maps from H to K. If H and K are finite-dimensional, then End(H,K) is a Hilbert

space endowed with the Hilbert-Schmidt inner product given by 〈A |B 〉End(H,K)

=

tr(A∗B) for A,B ∈ End(H,K). As usual, we write End(H) for End(H,H), and IH

for the identity map on H.


Proposition 1.1.9. Let (H, πH) and (K, πK) be two representations of a group G.

The map

πH,K : G −→ End(End(H,K))

g 7−→ πK(g)AπH(g−1)

defines a representation of G in End(H,K). The representation πH,K is unitary if

both πH and πKare.

Proof:

It is straightforward to show that πH,K is a group homomorphism, given that πH

and πK are. For A,B ∈ End(H,K), we have:

〈A |πH,K(g)B 〉End(H,K)

= tr(A∗πK(g)BπH(g−1)) = tr(πH(g−1)A∗πK(g)B)

= tr ((πH,K(g−1)A)∗B) = 〈πH,K(g−1)A |B 〉End(H,K)

By the uniqueness of the adjoint map, we have

(πH,K(g))∗ = πH,K(g−1)

Remark 1.1.10. If (H, πH) and (K, πK) are two representations of a group G, then

unless specified otherwise, the representation of G on End(H,K) will be taken to be

the one as given in Proposition 1.1.9.

1.2 G-equivariant maps

In this section, we provide all needed information about G-equivariant maps. The

first subsection contains the definitions and propositions. The second one presents a

list of examples of G-equivariant maps that are required for the thesis.


1.2.1 Basic definitions and results

Definition 1.2.1. [29, p.13] Let (H, πH) and (K, πK) be two representations of a

group G. A linear map α : H −→ K is said to be G-equivariant if

πK(g)α = απH(g)

for all g ∈ G.

The set of G-equivariant maps forms a vector space, denoted End(H,K)G, and

called the space of intertwining operators. The space End(H,H)G is abbreviated to

End(H)G. Following [3, p.67], we have

Definition 1.2.2. Let (H, πH) and (K, πK) be two representations of a group G.

We say that the two representations are G-equivalent if there exists a G-equivariant

isomorphism α : H −→ K. In such a case, the spaces H and K are called G-

equivalent, or G-isomorphic.

Proposition 1.2.3. Let G be a compact group. Then

1. Every representation of G in a Hilbert space is equivalent to a unitary represen-

tation [30, p.15].

2. Every representation of G in a Hilbert space is equivalent to a direct sum of

irreducible representations [37, p.155-157], and [3, p.68].

3. Every irreducible representation of G in a Banach space is finite-dimensional

[30, p.46].

Henceforth, we assume all vector spaces to be complex vector spaces of finite

dimension.

Proposition 1.2.4. Let (H, πH) and (K, πK) be two unitary representations of a

group G. The map Φ : End(H) −→ End(K) is G-equivariant if and only if

Φ(πH(g)Aπ∗H(g)) = πK(g)Φ(A)π∗K(g)


for all A ∈ End(H) and g ∈ G.

Proposition 1.2.5. For i = 1, 2, let (Hi, πHi) and (Ki, πKi) be representations of a

group G, and Φi : End(Hi) −→ End(Ki) be G-equivariant maps. The tensor product

and the direct sum of Φ1 and Φ2 are G-equivariant maps with respect to the actions

on the tensor product and the direct sum respectively. If Φ1 and Φ2 are composable

then their composition is also G-equivariant.

See Appendix A for definitions of the tensor product, the direct sum of two maps,

and the actions defined on their representation spaces.

Definition 1.2.6. Let H and K be Hilbert spaces, and α ∈ End(H,K). Conjugation

by α is the linear map

Adα : End(H) −→ End(K)

A 7−→ αAα∗

where α∗ is the conjugate map of α.


group G, and α : H −→ K be a G-equivariant map. Then

1. The image under α of a G-invariant subspace of H is G-invariant.

2. The conjugate of α is G-equivariant.

3. Conjugation by α is G-equivariant.

4. If α is an injective map then the image under α of any G-irreducible subspace

of H is G-irreducible.

Notation 1.2.8. For a subspace W of a Hilbert space H, let ιW denote the inclusion

map of W, and qW denote the orthogonal projection onto W .


Both maps are elements in End(H) (redefine ιW to be the identity map on W

and the zero on the orthogonal complement of W ). With this definition, the inclusion

map ιW is the conjugate of qW .

Remark 1.2.9. If (H, πH) is a unitary representation of a group G then the orthog-

onal complement of any G-invariant subspace is also G-invariant. For more details,

see [37, p.24], and [8, p.70].

Lemma 1.2.10. Let (H, πH) be a representation of a group G. The subspace W of

H is G-invariant if and only if qW (resp. ιW ) is a G-equivariant map.

Proof:

Suppose that W is G-invariant, and g ∈ G . Since any element h ∈ H can be

written as x+ y such that x ∈ W and y ∈ W⊥, then by the remark above, we have

qWπH(g)(h) = qW (πH(g)(x) + πH(g)(y)) = πH(g)(x) = πH(g)qW (h)

i.e. qW is G-equivariant. On other hand, if qW is G-equivariant then for w ∈ W and

g ∈ G, we have:

πH(g)(w) = πH(g)qW (w) = qWπH(g)(w) ∈ W.

Since ιW = q∗W , the equivalent follows for ιW .

Remark 1.2.11. Let H be a Hilbert space. If W is a subspace of H, given by the

orthogonal projection qW , then End(W ) is isomorphic to a subspace of End(H), given

by the projection AdqW . It follows that if (H, πH) is a representation of a group G with

a G-invariant subspace W then End(W ) is G-isomorphic to a G-invariant subspace

of End(H).

The following proposition is one of the main pillars in the study of group repre-

sentations. It is stated and proved in [9, p.7] for finite groups, but the method of the

proof is valid in the general case.


Proposition 1.2.12. (Schur’s Lemma) Let (H1, ρ1) and (H2, ρ2) be two irreducible

representations of a group G. If α : H1 −→ H2 is a G-equivariant map then either

α ≡ 0 or α is a G-isomorphism. In case of ρ1 = ρ2 and H1 = H2 then α is a multiple

of the identity.

Corollary 1.2.13. Let (H, πH) be a unitary representation of a group G. Any two

non isomorphic G-irreducible subspaces of H are mutually orthogonal.

Proof:

Let W1 and W2 be two G-invariant irreducible subspaces of H. By Lemma

1.2.10, the associated orthogonal projections qW1and qW2

are G-equivariant, hence

so is the map qW1q∗W2

= qW1ιW2

: W2 −→ W1. By Schur’s Lemma 1.2.12, the map

qW1q∗W2

= qW1qW2

is either the zero map or an isomorphism.

The proof of the next lemma is given in [9, p.7] for the case of finite groups (see

also [13, p.333 (21.40)]). However, the same statement and proof are valid for the

finite dimensional representation of compact group. Recall that we assumed that all

vector spaces are finite dimensional.

Lemma 1.2.14. Let (H, πH) be a representation of a compact group G. There exists

a decomposition

H = U⊕a11 ⊕ ...⊕ U⊕akk

where Ui are G-irreducible distinct subspaces. The decomposition of H into a direct

sum of the k factors is unique, as are the Ui occur and their multiplicities ai.

Definition 1.2.15. The number ai in Lemma 1.2.14, is called the multiplicity of the

subspace Ui.

Corollary 1.2.16. Let (H, πH) be a representation of a compact group G. Ifn⊕i=1

Ui and

m⊕j=1

Vj are two decompositions of H into G-irreducible subspaces of distinct dimensions


then m = n, and for each 1 ≤ i ≤ n, there exists a permutation σ ∈ Sn such that

Ui = Vσ(i).

Proof:

Fix 1 ≤ i ≤ n, by Lemma 1.2.14, m = n, and there exists a permutation σ ∈ Snsuch that Ui ' Vσ(i). Since all Vj have distinct dimensions then by Schur’s Lemma,

the subspaces Ui and Vj are orthogonal for each j 6= σ(i). Hence

Ui ⊆

⊕j 6=σ(i)

Vj

⊥ = Vσ(i).

Since Ui and Vσ(i) have the same dimension then Ui = Vσ(i).

Proposition 1.2.17. Let (H, πH) be a representation of a group G, such that H =m⊕i=1

Wi, where Wi are G-irreducible subspaces of H of multiplicity one. The space

End(H)G is a commutative algebra, that is spanned by the G-equivariant projections

on Wi : 1 ≤ i ≤ m.

Proof:

Let qi : H −→ Wi be the orthogonal projection onto the G-invariant subspace

Wi, and ιs : Ws −→ H be the inclusion map of Ws. By Lemma 1.2.10, the maps qi

and ιs are G-equivariant. Hence, for T ∈ End(H)G, the map qsTιi : Wi −→ Ws is

a G-equivariant map, intertwining the G-irreducible representations Ws and Wi. As

the multiplicity of each Wi is one, by Schur’s Lemma 1.2.12, we have

qsTιi =

0 s 6=i

λiIi s=i

where Ii is the identity map on Wi. As for h ∈ H, we can write h = w1 +w2 + ...+wm

where wi = qi(h), hence

T (h) = T (w1 + w2 + ...+ wm)


= λ1w1 + λ2w2 + ...+ λmwm

= λ1q1(h) + λ2q2(h) + ...+ λmqm(h)

=m∑i=1

λiqi(h).

To see that End(H)G is commutative, let T1 and T2 are two elements in End(H)G,

such that T1 =m∑i=1

λiqi and T2 =m∑s=1

µsqs. Since qi : 1 ≤ i ≤ m are mutually

orthogonal projections, then

T1T2 =m∑i=1

m∑i=1

λiµsqiqs =m∑i=1

λiµiqiqi

=m∑i=1

µiλiqi =m∑s=1

m∑i=1

µsλiqsqi = T2T1.

Proposition 1.2.18. Let (H, πH), (K, πK) and (E, πE) be unitary representations of

a group G, and α : H −→ K ⊗ E be a G-equivariant map. The map

T : E −→ End(H,K)

u 7−→ (IK ⊗ u∗)α

is G-equivariant, where u∗ denote the linear form on E given by u∗(z) = 〈u |z 〉E

.

Proof:

Let g ∈ G, and u ∈ E arbitrary elements. As πK(g)⊗ u∗ = πK(g)(IK ⊗ u∗) (check

on x⊗ y), we have

T (πE(g) (u)) = (IK ⊗ u∗π∗E(g))α = (IK ⊗ u∗π∗E(g)) ((πK(g)⊗ πE(g))απ∗H(g))

= (πK(g)⊗ u∗)απ∗H(g) = πK(g) (IK ⊗ u∗)απ∗H(g) = πK(g)T (u)π∗H(g)

i.e. T is G-equivariant.


Corollary 1.2.19. Let (H, πH), (K, πK), and (E, πE) be representations of a group

G such that πE is irreducible, and let α : H −→ K ⊗ E be a G-equivariant isometry.

If ej : 1 ≤ j ≤ dE is an orthonormal basis for E, then the set

Tj =(IK ⊗ e∗j

)α : 1 ≤ j ≤ dE ⊆ End(H,K)

satisfies

〈Tj1 |Tj2 〉End(H,K)= dH

dEδj1j2

for 1 ≤ j1, j2 ≤ dE.

Proof: Consider the G-equivariant map T : E −→ End(H,K), defined in Propo-

sition 1.2.18. By Schur’s Lemma 1.2.12, the map T ∗T : E −→ E is a multiple of the

identity on E. Thus, there exist λ ∈ C such that

〈T (u) |T (v)〉End(H,K)

= 〈u |T ∗T (v)〉E

= 〈u |λv 〉E

for any u, v ∈ E. Since

T (ej1) = Tj1 , and T (ej2) = Tj2 for 0 ≤ j1, j2 ≤ dE.

then

〈Tj1 |Tj2 〉End(H,K)= 〈T (ej1) |T (ej2)〉

End(H,K)

= λ 〈ej1 |ej2 〉E = λδj1j2

As Tj =(IK ⊗ e∗j

)α, the map α is equal to

dE∑j=1

Tj ⊗ ej. Let fi : 1 ≤ i ≤ dH be an

orthonormal basis of H, then

dH =dH∑i=1

‖α(fi)‖2 =dH∑i=1

∥∥∥∥ dE∑j=1

Tj(fi)⊗ ej∥∥∥∥2

=

dH∑i=1

dE∑j=1

‖Tj(fi)‖2 =

dE∑j=1

‖Tj‖2 =

dE∑j=1

λ = dEλ

i.e. λ = dHdE

.


1.2.2 Examples of G-equivariant maps

In this subsection, we provide examples, and standard constructions of G-equivariant

maps. Recall that if (H, πH) and (K, πK) are representations of a group G, then

πH ⊗ πK is a representation of G in the space H ⊗ K. The new representation is

defined naturally by linearly extending πH ⊗ πK(g)(h⊗ k) = πH(g) (h) ⊗ πK(g) (k) for

g ∈ G, and h, k ∈ H,K. For two matrices A = (aij) and B = (blk), the Kronecker

product of A and B is defined to be A⊗B = (Ablk).

I. The partial trace:

Definition 1.2.20. Let H and K be Hilbert spaces. The linear map

TrH : End(H ⊗K) −→ End(K)

A⊗B 7−→ tr(A)B

defined for A⊗B∈End(H ⊗K), and extended by linearity is called the partial trace

over H.

In similar way, we define the partial trace over K by taking the trace over the

second component of A⊗B.

Lemma 1.2.21. Let (H, πH) and (K, πK) be two unitary representations of a group

G. The partial trace over H is a G-equivariant map.

Proof:

Let g ∈ G, and A1 ⊗ A2 ∈ End(H ⊗K). Then

TrH ((πH(g)⊗ πK(g)) (A1 ⊗ A2) (π∗H(g)⊗ π∗K(g)))

= TrH(πH(g)A1π∗H(g)⊗ πK(g)A2π

∗K(g))


= tr(πH(g)A1π∗H(g))πK(g)A2π

∗K(g)

= tr(A1)πK(g)A2π∗K(g) = πK(g)TrH(A1 ⊗ A2)π∗K(g)

II. The flipping map (swap map):

Our second example of G-equivariant maps is defined on the tensor product of two

vector spaces H and K. It is called the flipping map (swap map), and denoted by

flipHK.

Definition 1.2.22. Let H and K be Hilbert spaces. We define the linear map

flipHK : H ⊗K −→ K ⊗H

via

h⊗ k 7−→ k ⊗ h

on the set h⊗ k : h ∈ H, k ∈ K and then extended linearly.

Lemma 1.2.23. Let (H, πH) and (K, πK) be two representations of a group G. The

map flipHK is a unitary G-equivariant map satisfying (flipHK)∗ = flipKH and

TrK(flipHK AflipK

H) = TrK(A)

for A ∈ End(H ⊗K).

Proof:

Let g ∈ G. For h⊗ k ∈ H ⊗K, we have

(πK(g)⊗ πH(g)) flipHK(h⊗ k) = (πK(g)⊗ πH(g)) (k ⊗ h)


= πK(g)(k)⊗ πH(g)(h) = flipHK(πH(g)(h)⊗ πK(g)(k))

= flipHK (πH(g)⊗ πK(g)) (h⊗ k)

G-equivariance follows by linearity. Since for (h ⊗ k) ∈ H ⊗K, ‖h⊗ k‖ = ‖k ⊗ h‖,

flipHK is a unitary map such that (flipHK)∗ = (flipHK)−1 = flipKH .

To prove the second assertion, let B1 ∈ End(H) and B2 ∈ End(K). By direct

computations on arbitrary element k ⊗ h, we have

flipHK (B1 ⊗B2) flipKH = B2 ⊗B1

Hence,

TrK(flipHK (B1 ⊗B2) (flipHK)∗) = tr(B2)B1 = TrK(B1 ⊗B2).

The result follows by linearity.

III. The map Vec:

Recall that xy∗ : x ∈ K, y ∈ H forms a set of generators of End(H,K), where

y∗ denotes the linear form on H given by y∗(z) = 〈y |z 〉H

, and xy∗ denotes the map

xy∗(z) = 〈y |z 〉Hx for any z ∈ H. The following is a reformulation of the definition

of the map vec in [41, p.23].

Definition 1.2.24. Let H and K be Hilbert spaces, let Vec : End(H,K) −→ K ⊗H

be the linear map defined on the elements xy∗ of End(H,K) via

xy∗ 7−→ x⊗ y

extended linearly.

The map Vec represents any element in End(H,K) as a vector in the tensor

product space K ⊗H.


Lemma 1.2.25. Let (H, πH) and (K, πK) be two representations of a group G. By

considering the conjugate representation on H, the map

Vec : End(H,K) −→ K ⊗H

is a G-equivariant unitary map.

Proof:

Let ei : 1 ≤ i ≤ dH and e′j : 1 ≤ j ≤ dK be orthonormal bases for H and K

respectively, hence the set Eij = eie′∗j : 1 ≤ i ≤ dH, 1 ≤ j ≤ dK is an orthonormal

basis for End(H,K). For g ∈ G and Eij ∈ End(H,K), we have

Vec(πK(g)Eijπ∗H(g)) = Vec

(πK(g)ei

(πH(g)e′j

)∗)= πK(g)ei ⊗ πH(g)e′j

= (πK ⊗ πH) (g)(ei ⊗ e′j) = (πK ⊗ πH) (g)Vec(Eij).

The G-equivariant follows by linearity of the map Vec. Clearly Vec is a linear bijection

which is unitary since

〈A |B 〉End(H,K)

= 〈Vec(A) |Vec(B)〉K⊗H

for A,B ∈ End(H,K).

IV. The Choi-Jamiolkowski map:

The following is an equivalent definition of the Choi-Jamiolkowski map in [41, p.49];

we show this equivalence in Lemma 1.2.30.

Definition 1.2.26. Let H and K be Hilbert spaces. The linear map

C : End(End(H), End(K)) −→ End(K ⊗H)

AB∗ 7−→ A⊗B


defined for A ∈ End(K) and B ∈ End(H) and extended linearly is called the Choi-

Jamiolkowski map. The map B∗ : End(H) −→ C is given by B∗X = 〈B |X 〉End(H)

for X ∈ End(H).

Lemma 1.2.27. Let (H, πH) and (K, πK) be two representations of a group G. The

natural isomorphism T : End(K)⊗End(H) −→ End(K⊗H) defined by taking A⊗B

to T (A⊗B)(k ⊗ h) = A(k)⊗B(h) and extending linearly is a G-equivariant map.

Recall that for finite-dimensional Hilbert spaces H and K, the spaces

End(End(H), End(K)), End(K)⊗End(H)∗, End(K)⊗End(H), and End(K ⊗H)

are all algebraically isomorphic. The spaces End(H) and End(H) are equal. See

Appendix A, for more details.

Remark 1.2.28. The Choi-Jamiolkowski map is the composition of the maps

Vec : End(End(H), End(K)) −→ End(K)⊗ End(H) = End(K)⊗ End(H)

with the natural isomorphism T : End(K) ⊗ End(H) −→ End(K ⊗ H), defined in

Lemma 1.2.27.

By Lemma 1.2.25, and the remark above, we have:

Corollary 1.2.29. Let (H, πH) and (K, πK) be two representations of a group G. The

Choi-Jamiolkowski map is unitarily G-equivariant.

The Choi-Jamiolkowski map assigns to each Φ ∈ End(End(H), End(K)) a

unique matrix C(Φ) ∈ End(K ⊗H), called the Choi matrix of Φ. The next lemma

shows the equivalency of Definition 1.2.26, and the original definition of Choi-Jamiolkowski

map in [23, p.276], and [41, p.49].

Lemma 1.2.30. Let H and K be Hilbert spaces. For Φ ∈ End(End(H), End(K)),

the Choi matrix of Φ is given by

C(Φ) =∑i,j

Φ(Eij)⊗ Eij

where Eij : 0 ≤ i, j ≤ dH is the standard orthonormal basis for End(H).


Proof:

It suffices to show the equality for Φ = AB∗, where A ∈ End(K) and B ∈

End(H).

Let B =dH∑k,l

λklEkl, then

dH∑ij

AB∗(Eij)⊗ Eij =

dH∑ij

A 〈B |Eij 〉 ⊗ Eij =

dH∑ij

Aλij ⊗ Eij

= A⊗dH∑ij

λijEij = A⊗dH∑ij

λijEij

= A⊗B = C(AB∗)

Next we determine the conditions on the Choi matrix of a linear map that are

equivalent to G-equivariance. The idea of the proof is taken from [7, p.5].

Lemma 1.2.31. Let H and K be Hilbert spaces. For Φ ∈ End(End(H), End(K)),

αK ∈ End(K), and αH ∈ End(H) we have:

1. C(AdαK Φ) = (αK ⊗ IH)C(Φ)(αK ⊗ IH)∗.

2. C(Φ AdαH ) = (IK ⊗ αH)∗C(Φ)(IK ⊗ αH).

Proof:

Let Eij : 0 ≤ i, j ≤ dH be the standard orthonormal basis for End(H). By

Lemma 1.2.30, we have

(αK ⊗ IH)C(Φ)(αK ⊗ IH)∗ = (αK ⊗ IH)

(∑i,j

Φ(Eij)⊗ Eij

)(αK ⊗ IH)∗


=∑i,j

(αK ⊗ IH) (Φ(Eij)⊗ Eij) (αK ⊗ IH)∗

=∑i,j

(αKΦ(Eij)α∗K)⊗ (IHEijI

∗H)

=∑i,j

(AdαK Φ(Eij))⊗ Eij = C(AdαK Φ)

establishing the first equality. For the second equality, let A ∈ End(K) and B ∈

End(H), it suffices to show that the equality holds for Φ = AB∗. Let D ∈ End(H),

we have:

AB∗ AdαH (D) = AB∗ (αHDα∗H) = A 〈B | αHDα∗H〉End(H)

= tr(αHDα∗HB∗)A = tr(D(α∗HBαH)∗)A

= A 〈αHBα∗H | D〉End(H)= A(α∗HBαH)∗(D)

that is,

AB∗ AdαH = A(α∗HBαH)∗

consequently,

C(AB∗ AdαH ) = C(A(α∗HBαH)∗) = A⊗ α∗HBαH

= IKAIK ⊗ α∗HBαH = (IK ⊗ αH)∗(A⊗B

)(IK ⊗ αH)

= (IK ⊗ αH)∗C(AB∗)(IK ⊗ αH).

Recall that for a group G, and a Hilbert space H, the set End(H)G denotes the

set of G-equivariant maps on H.


group G. A linear map Φ : End(H) −→ End(K) is G-equivariant if and only if

C(Φ) ∈ End(K ⊗H)G.


Proof:

By injectivity of the Choi-Jamiolkowski map, Φ is G-equivariant if and only if

C(AdπK(g) Φ) = C(Φ AdπH(g)) ∀g ∈ G

By Lemma 1.2.31, this holds if and only if

(πK(g)⊗ IH(g))C(Φ)(πK(g)⊗ IH(g))∗ = (IK(g)⊗ πH(g))∗C(Φ)(IK(g)⊗ πH(g)) ∀g ∈ G

⇐⇒ (πK(g)⊗ πH(g))C(Φ) = C(Φ)(πK(g)⊗ πH(g)) ∀g ∈ G

⇐⇒ C(Φ) ∈ End(K ⊗H)G.

Chapter 2

Representations of SU(2)

According to the Clebsch-Gordan Decomposition [3, p.87], if H and K are two SU(2)-

irreducible subspaces, then the SU(2)-space K ⊗ E is isomorphic to⊕i

Hi where Hi

is SU(2)-irreducible subspace with multiplicity one. For each i, the inclusion map

αi : Hi −→ K ⊗ E is SU(2)-equivariant. Our main goal in this chapter is to find

an explicit formula for the map αi. In the first section, we review the irreducible

representations of SU(2), and for an SU(2)-irreducible space H, we find an SU(2)-

equivariant unitary map from H onto H. In Section 2.2, we state the Clebsch-

Gordan expansion theorem. We get our main result of this chapter in Section 2.3,

by constructing an SU(2)-equivariant isometry defined on the representation space

of the SU(2)-irreducible representation. We end this chapter with an application of

our results, in Section 2.4. The proofs of some Lemmas and corollaries in this chapter

are purely technical calculations, and are deferred to Appendix B.

The main results of this chapter:

• Constructing an SU(2)-equivariant unitary map between any SU(2)-irreducible

space, and its corresponding conjugate space (Proposition 2.1.6).

27

2. Representations of SU(2) 28

• Constructing an SU(2)-equivariant isometry defined on the representation space

of the SU(2)-irreducible representation (Proposition 2.3.3, and Proposition 2.3.5).

• For an SU(2)-irreducible subspace H, we give explicit formulae for the pro-

jections of End(H) into its SU(2)-irreducible invariant subspaces (Proposition

2.4.2).

2.1 The irreducible representations of SU(2)

For m ∈ N, let Pm denote the space of homogeneous polynomials, with complex

coefficients, of degree m in the two variables x1, x2. It is a complex vector space of

dimension m + 1 with a basisxi1x

m−i2 : 0 ≤ i ≤ m

. By convention, we denote by

P−1 the zero vector space. Recall that

SU(2) =

a b

−b a

: a, b ∈ C, |a|2 + |b|2 = 1

Definition 2.1.1. For m ∈ N, define ρm : SU(2) −→ End(Pm) by

(ρm(g)f) (x1,x2) = f ((x1,x2)g) = f(ax1 − bx2, bx1 + ax2)

for f ∈ Pm and g ∈ SU(2).

The next proposition summarizes results in [3, p. 85-86], [38, p.181], and [39,

p.276-279].

Proposition 2.1.2.

1. For m ∈ N, ρm is a unitary representation of SU(2) with respect to the inner

product on Pm given by

⟨xl1x

m−l2 , xk1x

m−k2

⟩Pm

= l! (m− l)! δlk


2. The set ρm : m ∈ N constitutes the full list of the irreducible representations

of SU(2).

To facilitate the computations, we choose the orthonormal basis for Pm given

by the polynomialsfml = almx

l1x

m−l2 : 0 ≤ l ≤ m

where alm =

1√l!(m−l)!

; this basis is

called canonical [39, p.280]. Throughout this thesis, we utilize the following definition

and notation:

Definition 2.1.3. For m ∈ N, the set fml : 0 ≤ l ≤ m is called the standard basis

for the SU(2)-irreducible space Pm. The corresponding standard basis for End(Pm) is

Elk = fml−1fm∗k−1 : 1 ≤ l, k ≤ m+ 1.

Remarks 2.1.4. Let g = a b

−b a

∈ SU(2). For m ∈ N, we have

1. The map ρm(g) is given on the standard basis for Pm by

ρm(g)(fml ) = alm(ax1 − bx2)l(bx1 + ax2)m−l

In particular, for g0 = 0 1

−1 0

∈ SU(2), we have

ρm(g0) (fml ) = (−1) lfmm−l

and

ρm(g∗0) (fml ) = (−1) m−lfmm−l

2. The map ρm(g) belongs to both End(Pm) and End(Pm). For if g ∈ SU(2) then

g∈ SU(2), and ρm(g) is a unitary map on Pm. Thus ρm(g) = ρm(g) is a unitary

map on Pm, where Pmis the conjugate space for Pm.

The element 0 1

−1 0

plays a special role in constructing an SU(2)-equivariant

unitary map from Pm onto Pm; we will denote this element by g0.


Definition 2.1.5. For m ∈ N, define the endomorphisms

1. Θm : Pm −→ Pm by Θm

(m∑l=0

λlfml

)=

m∑l=0

λl · fml , where · is the multiplication in

Pm.

2. Jm : Pm −→ Pm by Jm = Θmρm(g0).

Proposition 2.1.6. For m ∈ N,

1. Θm is a unitary map that satisfies ρm(g)Θm = Θmρm(g) for any g ∈ SU(2).

2. Jm is an SU(2)-equivariant unitary map from Pm onto Pm.

Proof:

The map Θm takes the orthonormal basis fml : 0 ≤ l ≤ m for Pm to itself; as it

is a basis for Pm, hence Θm is a unitary map. Let g ∈ SU(2), by Remark 2.1.4,

Θm ρm(g)(fml ) = Θm

(alm(ax1 − bx2)l(bx1 + ax2)m−l

)= alm(a · x1 − b.x2)l(b.x1 + a.x2)m−l

= ρm(g) (Θm(fml )) = ρm(g) Θm(fml ).

For the second statement, as Jm is a composition of two unitary maps, it is unitary.

Since

g0g = gg0 ∀g ∈ SU(2)

we have,

Jmρm(g) = Θmρm(g0)ρm(g) = Θmρm(g)ρm(g0)

= ρm(g)Θmρm(g0) = ρm(g)Jm.


Remark 2.1.7. For m ∈ N and 0 ≤ l ≤ m, we have:

Jm(fml ) = (−1)lfmm−l, J∗m(fml ) = (−1)m−lfmm−l

where fml is the basis element for Pm.

For m,n ∈ N, we fixed our choice for a basis for Pm ⊗ Pn to be

fml ⊗ fnj : 0 ≤ l ≤ m, 0 ≤ j ≤ n

where writing the basis in this form means that we choose the order to be in the form

fm0 ⊗fn0 , fm1 ⊗fn0 , ...fmm⊗fn0 , fm0 ⊗fn1 , fm1 ⊗fn1 , ......fmm⊗fn1 , ....., fm0 ⊗fnn , fm1 ⊗fnn , .....fmm⊗fnn

In this thesis, we call this basis the standard basis for Pm ⊗ Pn. Recall the flip map

in Definition 1.2.22. By direct computations on the elements of the standard basis

for Pm ⊗ Pn, we obtain

Proposition 2.1.8. For m,n ∈ N, the map

flipPmPn (Jm ⊗ IPn) : Pm ⊗ Pn −→ Pn ⊗ Pm

is an SU(2)-equivariant isomorphism, that satisfies

flipPmPn (Jm ⊗ IPn) = (IPn ⊗ Jm)flipPmPn

2.2 Clebsch-Gordan expansion

For m,n ∈ N, let ρm and ρn be the irreducible representations of SU(2) with cor-

responding SU(2)-spaces Pm and Pn. One can construct a new representation of

SU(2) by taking the tensor product of the two representations, which is not neces-

sarily irreducible. In this section, we build polynomial operators on the SU(2)-space

Pm⊗Pn. To obtain a concrete representation of Pm⊗Pn, let x = (x1, x2), y = (y1, y2),

Pm := Pm(x), and Pn := Pn(y). We embed the tensor product Pm(x) ⊗ Pn(y) into

C[x, y] as follows:


Define the map : Pm(x)× Pn(y) −→ C[x, y] by (f(x), g(y)) 7−→ f(x)g(y). It is a

bilinear map, hence extends to a linear T : Pm(x)⊗ Pn(y) −→ C[x, y] taking

f(x)⊗ g(y) to f(x)g(y). Let Pm,n denote the vector space of polynomials in x and y

of bi-degree (m,n) (homogeneous polynomials of degree m in x = (x1, x2) and of

degree n in y = (y1, y2)). The space Pm,n has a basis consisting of

xs1xm−s2 yt1yn−t2 =

1

ams ant

T (fms ⊗ fnt ) : 0 ≤ s ≤ m, 0 ≤ t ≤ n

Since the map T takes a basis for Pm ⊗ Pn to a basis in Pm,n, it is an isomorphism.

Henceforth, we will use Pm,n as a concrete representation of Pm ⊗ Pn.

Remark 2.2.1. Using the identification between Pm⊗Pn and Pm,n above, and Remark

2.1.4, we have

ρm(g)⊗ ρn(g)f(x, y) = f(ax1 − bx2, bx1 + ax2, ay1 − by2, by1 + ay2)

where f(x, y) := f(x1, x2, y1, y2) ∈ Pm ⊗ Pn, and g ∈ SU(2).

Recall that C[x1, x2, y1, y2,∂∂x1, ∂∂x2, ∂∂y1, ∂∂y2

] is a non-commutative algebra of poly-

nomial differential operators which acts on C[x1, x2, y1, y2]. The multiplication is the

composition of operators.

Definition 2.2.2. For m,n ∈ N, define the following maps on Pm ⊗ Pn

∆xy : Pm ⊗ Pn −→ Pm+1 ⊗ Pn−1

f(x, y) 7−→(x1

∂∂y1

+ x2∂∂y2

)f(x, y)

∆yx : Pm ⊗ Pn −→ Pm−1 ⊗ Pn+1

f(x, y) 7−→(y1

∂∂x1

+ y2∂∂x2

)f(x, y)


Γxy : Pm ⊗ Pn −→ Pm+1 ⊗ Pn+1

f(x, y) 7−→ (x1y2 − y1x2) f(x, y)

Ωxy : Pm ⊗ Pn −→ Pm−1 ⊗ Pn−1

f(x, y) 7−→(

∂∂x1

∂∂y2− ∂

∂x2

∂∂y1

)f(x, y)

for f(x, y) ∈ Pm ⊗ Pn.

The statement of the next lemma is mentioned in [29, p.47]. The proof is purely

computational, and is provided in Appendix B.

Lemma 2.2.3. The operators ∆xy, ∆yx, Γxy, and Ωxy are SU(2)-equivariant, and

satisfy

∆∗xy = ∆yx, Γ∗xy = Ωxy

Theorem 2.2.4. (Clebsch-Gordan expansion)[29, p.46]. Let m,n ∈ N. For a

polynomial f(x, y) ∈ Pm ⊗ Pn, we have

f(x, y) =

minm,n∑h=0

cm,n,h Γhxy∆n−hyx ∆n−h

xy Ωhxy(f(x, y))

where the coefficients cm,n,h are determined by induction as follows: cm,0,0 = 1. For

n ≥ 1 and 0 ≤ h ≤ minm,n.

cm,n,h =

1

(m+1)ncm+1,n−1,h h=0

1(m+1)n

[cm−1,n−1,h−1 + cm+1,n−1,h] 0<h<n

1(m+1)n

cm−1,n−1,h−1 h=n


2.3 An SU(2)-equivariant isometry on the repre-

sentation space of SU(2)

In this section, we use Lemma 2.2.3 and Theorem 2.2.4 to define an SU(2)-equivariant

isometry αm,n,h on the SU(2)-irreducible space Pm+n−2h, where m,n, h ∈ N with

0 ≤ h ≤ minm,n. We also give the matrix coefficients of αm,n,h with respect

to the standard basis for Pm+n−2h.

2.3.1 The isometry αm,n,h

The proof of the next proposition is in [3, p.87].

Proposition 2.3.1. (Clebsch-Gordan decomposition) For m,n ∈ N, let ρm and

ρn be the irreducible representations of SU(2) with corresponding SU(2)-spaces Pm

and Pn. Then

ρm ⊗ ρn =

minm,n⊕h=0

ρm+n−2h

Consequently, the SU(2)-space Pm ⊗ Pn is isomorphic tominm,n⊕h=0

Pm+n−2h.

Definition 2.3.2. For m,n, h ∈ N with 0 ≤ h ≤ min m,n, let

αm,n,h : Pm+n−2h −→ Pm⊗Pn

be the linear map defined by

αm,n,h(f(x1, x2)) =√cm,n,h Γhxy∆

n−hyx (f(x1, x2))

where f(x1, x2) is homogeneous polynomial in x1, x2 of degree m+ n− 2h.

By Lemma 2.2.3, and the fact that a composition of two G-equivariant maps is

G-equivariant, we have:


Proposition 2.3.3. For m,n, h ∈ N with 0 ≤ h ≤ minm,n, the map αm,n,h is an

SU(2)-equivariant map.

As a result of Theorem 2.2.4, and Lemma 2.2.3, we get

Lemma 2.3.4. Let m,n, h ∈ N with 0 ≤ h ≤ minm,n.

1. The conjugate map of αm,n,h is given by

α∗m,n,h : Pm ⊗ Pn −→ Pm+n−2h

α∗m,n,h (g(x1, x2, y1, y2)) =√cm,n,h ∆n−h

xy Ωhxy(g(x1, x2, y1, y2))

where g(x1, x2, y1, y2) is a homogeneous polynomial of degree m in x1, x2, and

homogeneous of degree n in y1, y2.

2.minm,n∑h=0

αm,n,hα∗m,n,h = IPm⊗Pn.

Proposition 2.3.5. For m,n, h ∈ N with 0 ≤ h ≤ minm,n, the map αm,n,h is an

isometry.

Proof:

For 0 ≤ h, s ≤ minm,n, the map α∗m,n,hαm,n,s : Pm+n−2s −→ Pm+n−2h is an

SU(2)-equivariant map. By Schur’s Lemma 1.2.12, we have

α∗m,n,hαm,n,s =

0 if h6=s

λIPm+n−2hif h=s

(2.3.1)

for some non-negative integer λ. It remains to show that λ = 1. By Lemma 2.3.4,

the mapminm,n∑

s=0

αm,n,sα∗m,n,s is the identity map on the space Pm⊗Pn. Thus, for any

0 ≤ h ≤ minm,n, we have

α∗m,n,h = α∗m,n,h IPm⊗Pn = α∗m,n,h

minm,n∑s=0

αm,n,sα∗m,n,s =

minm,n∑s=0

α∗m,n,hαm,n,sα∗m,n,s


By Equation (2.3.1) above, we get

α∗m,n,h = α∗m,n,hαm,n,hα∗m,n,h = λα∗m,n,h

Since αm,n,h 6= 0 (αm,n,h(xm+n−2h

1 ) 6= 0), and∥∥α∗m,n,h∥∥ = ‖αm,n,h‖, then

α∗m,n,h 6= 0 which gives λ = 1.

Corollary 2.3.6. Let m,n ∈ N. The SU(2)-space

Pm ⊗ Pn =

minm,n⊕h=0

Wm+n−2h

where Wm+n−2h ' Pm+n−2h, 0 ≤ h ≤ minm,n are mutually orthogonal SU(2)-

subspaces, such that Wm+n−2h is the range of the orthogonal projection αm,n,hα∗m,n,h.

This decomposition into SU(2)-irreducible subspaces is unique up to permutation.

Proof:

Let 0 ≤ h ≤ minm,n. By Proposition 2.3.5, the map αm,n,hα∗m,n,h is an

orthogonal projection. Let Wm+n−2h = αm,n,hα∗m,n,h(Pm⊗Pn). By surjectivity of α∗m,n,h,

the equivariance of αm,n,h, and the irreducibility of Pm+n−2h, the subspace Wm+n−2h =

αm,n,h(Pm+n−2h) is an SU(2)-irreducible subspace of Pm ⊗ Pn that is isomorphic to

Pm+n−2h. The mutual orthogonality of the subspaces Wm+n−2h follows by Corollary

1.2.13. Henceminm,n⊕h=0

Wm+n−2h ⊆ Pm ⊗ Pn

By comparing the dimensions, we get the equality Pm ⊗ Pn =minm,n⊕h=0

Wm+n−2h. The

uniqueness follows by Corollary 1.2.16.

A similar result can be stated for the space Pm⊗P n. We need the following

lemma, whose proof is straightforward using Proposition 2.1.6. Recall that P n is an

SU(2)-irreducible space under the contragredient representation ρn.


Lemma 2.3.7. Let m,n ∈ N.

1. The map IPm ⊗ Jn : Pm ⊗ Pn −→ Pm ⊗ P n is an SU(2)-equivariant unitary

isomorphism whose inverse is IPm ⊗ J∗n .

2. The map ηm,n,h = (IPm ⊗ Jn)αm,n,h is an SU(2)-equivariant isometry from

Pm+n−2h into Pm ⊗ P n.

Corollary 2.3.8. Let m,n ∈ N. The SU(2)-space

Pm ⊗ P n =

minm,n⊕h=0

Vm+n−2h

where Vm+n−2h ' Pm+n−2h, 0 ≤ h ≤ minm,n are mutually orthogonal SU(2)-

subspaces, such that Vm+n−2h is the range of the orthogonal projection ηm,n,hη∗m,n,h.

This decomposition into SU(2)-irreducible subspaces is unique up to permutation.

2.3.2 The matrix coefficients of αm,n,h

In this section, we give the matrix coefficients of the isometry αm,n,h with respect

to the standard basis for Pm+n−2h. The proof of Lemma 2.3.11 below is a direct

computation, and is deferred to Appendix B. To simplify our notation, we state the

following definition.

Definition 2.3.9. For a subset A of a space X, the indicator function χA : X −→

0, 1 is defined by

χA(x) =

1 if x ∈ A

0 if x /∈ A

For the rest of the thesis, we will systematically use the following notation without

further mention. For quick reference, Appendix C contains the list of notation and

equation that are used in the thesis.

Notation 2.3.10. For m,n, h, i, j ∈ N with 0 ≤ h ≤ minm,n, 0 ≤ i ≤ m+n−2h,

and 0 ≤ j ≤ n, we define


• B(i) := j : max0,−m+ i+ h ≤ j ≤ mini+ h, n.

• lij := i− j + h.

Lemma 2.3.11. Let m,n, h ∈ N with 0 ≤ h ≤ minm,n, and r = m + n − 2h.

Then

1. αm,n,h (f ri ) =h∑s=0

mini+s, n−h+s∑j=maxs,−m+i+h+s

βm,n,hi,s,j fmlij ⊗ fnj

2. α∗m,n,h(fml ⊗ fnj

)=

(minh,j,m−l∑

s=max0,h−l,h+j−nβm,n,hl+j−h,s,j

)f rl+j−h · χ[0,r](l+j−h)

where

βm,n,hi,s,j = (−1)s(hs

)(n−hj−s

)(m−hi−j+s

)(m−h)!

√cm,n,hr! m! n!

(ri) ( mi−j+h) (nj)

and fkt : 0 ≤ t ≤ k is the standard basis for Pk, where k ∈ m,n, r.

Remark 2.3.12. The vectors fmlij and f rl+j−h, in Lemma 2.3.11, are indeed elements

of the basis for Pm, and Pr respectively. This follows since

maxs,−m+ i+ h+ s ≤ j ≤ mini+ s, n− h+ s

implies

0 ≤ max0, i− n+ h ≤ lij ≤ min i+ h,m ≤ m

and the summinh,j,m−l∑

s=max0,h−l,h+j−nβm,n,hl+j−h,s,j is nonempty if and only if

max0, h− l, h+ j − n ≤ minh, j,m− l

which implies 0 ≤ l + j − h ≤ r.

The following corollary gives more compact forms of the formulae in Lemma

2.3.11.


Corollary 2.3.13. For m,n, h ∈ N with 0 ≤ h ≤ minm,n, let r = m + n − 2h.

The values of the maps αm,n,h and α∗m,n,h on the elements of the standard bases for

Pr and Pm ⊗ Pn are given by

αm,n,h (f ri ) =∑j∈B(i)

εji (m,n,h) fmlij ⊗ fn

j

and

α∗m,n,h(fml ⊗ fnj

)= εjl+j−h(m,n,h)f rl+j−h · χ[0,r](l+j−h)

where εjt (m,n,h) =minh,j,j+m−t−h∑s=max0,j−t,j+h−n

βm,n,ht,s,j .

Proof:

For the first equality, let

A = (s, j) : 0 ≤ s ≤ h, maxs,−m+ i+ h+ s ≤ j ≤ mini + s, n − h + s, and

Aj = s : (s, j) ∈ A then

A = ∪j(s, j) : s ∈ Aj

where the index j ranges from max0,−m+ i+ h at s = 0 to mini + h, n at

s = h.

i.e

Aj = s : (s, j) ∈ A = s : 0 ≤ s ≤ h, j −mini, n− h ≤ s ≤ j −max0,−m+ i+ h

= s : max0, j −mini, n− h ≤ s ≤ minh, j −max0,−m+ i+ h

= s : max0, j − i, j + h− n ≤ s ≤ minh, j, j +m− i− h

By Lemma 2.3.11, we have

αm,n,h(fr

i ) =∑

(s,j)∈A

βm,n,hi,s,j fmlij ⊗ f

n

j =

mini+h, n∑j=max0,−m+i+h

∑s∈Aj


n

j

=


minh,j,j+m−i−h∑s=max0,j−i,j+h−n


n

j =∑j∈B(i)

εji (m,n,h) fmlij ⊗ fn

j .


The second equality follows by Lemma 2.3.11.

For the rest of the thesis, when m,n, h are clear from context, we will abbreviate

εji (m,n,h) to εji .

Remarks 2.3.14. Let m,n, h ∈ N, with 0 ≤ h ≤ minm,n, and r = m+ n− 2h.

1. For 0 ≤ i ≤ r, we have

(a) The set B(i) is non-empty, as max0,−m+ i+ h ≤ mini+ h, n.

(b) B(r − i) = n−B(i).

2. If Bm,n,h(i) denotes the set B(i) associated to αm,n,h, and Bn,m,h(i) denotes the

set B(i) associated to αn,m,h, then

j ∈ Bm,n,h(i) if and only if lij ∈ Bn,m,h(i)

3. The map αm,n,h can be written as

αm,n,h =r∑i=0

∑j∈B(i)

εji

(fmlij ⊗ f

n

j

)f r∗

i

4. The map ηm,n,h in Lemma 2.3.7 is given on the basis elements for Pr by

ηm,n,h (f ri ) =∑j∈B(i)

(−1)jεji (m,n,h) fmlij ⊗ fn

n−j

The following corollary, whose proof is in Appendix B, contains basic relations

for the coefficients εji .

Corollary 2.3.15. For m,n, h ∈ N with 0 ≤ h ≤ minm,n, let r = m + n − 2h.

The matrix coefficients εji := εji (m,n,h) of the isometry αm,n,h satisfy

1. εji = (−1)hεn−jr−i for 0 ≤ i ≤ r and j ∈ B(i).


2. εi+hi = βm,n,hi,h,h+i for i ≤ n− h.

3. εni = βm,n,hi,h,n = (−1)h∣∣∣βm,n,hi,h,n

∣∣∣ 6= 0 for n− h ≤ i ≤ r.

4. εj0 = βm,n,h0,j,j = (−1)j

∣∣∣βm,n,h0,j,j

∣∣∣ 6= 0 for j ∈ B(0).

5. ε0i = βm,n,hi,0,0 6= 0 for 0 ≤ i ≤ m− h.

6. εji (m,n,n) = βm,n,ni,j,j , and εji (m,n,0) = βm,n,0i,0,j .

Corollary 2.3.16. Let m,n, h ∈ N with 0 ≤ h ≤ minm,n, and r = m+ n− 2h.

1. The coefficient cm,n,h in the Clebsch-Gordan expansion (Theorem 2.2.4), is given

by

cm,n,h =((m−h)!)2

r! m! n!

(h∑k=0

(hk

)2

( mh−k) (nk)

)Consequently,

βm,n,hi,s,j =(−1)s

(hs

) (n−hj−s

) (m−hi−j+s

)√√√√(ri

) (m

i−j+h

) (nj

)( h∑k=0

(hk

)2

( mh−k) (nk)

)

2. For 0 ≤ i ≤ r , and j ∈ B(i), we have

εji (m,n,h) = (−1)hεliji (n,m,h)

Proof:

By Corollary 2.3.13, we have αm,n,h(fr0 ) =

h∑j=0

βm,n,h0,j,j fml0j ⊗ fnj . Since

fml0j ⊗ f

nj

are orthonormal set, and αm,n,h is isometry, then 1 =

h∑j=0

(βm,n,h0,j,j

)2

. As

βm,n,h0,j,j = (−1)j√cm,n,h

(hj

)(m−h)!

√r! m! n!

( mh−j) (nj)

,


the first statement follows. For the second one, by using the formula for εji in Corollary

2.3.13, we have

εliji (n,m,h) =

minh,lij ,lij−i−h+n∑t=max0,lij−i,lij+h−m

βn,m,hi,t,lij=

minh,i−j+h,n−j∑t=max0,h−j,i+2h−j−m

βn,m,hi,t,lij

By Remarks 2.3.14 (2), and since βn,m,hi,h−s,lij = (−1)hβm,n,hi,s,j for any 0 ≤ s ≤ h, we have

εliji (n,m,h) =

minh,i−j+h,n−j∑t=max0,h−j,i+2h−j−m

βn,m,hi,t,lij=

minh,j,j−i+m−h∑s=max0,j−i,j+h−n

βn,m,hi,h−s,lij

=

minh,j,j−i+m−h∑s=max0,j−i,j+h−n

(−1)hβm,n,hi,s,j = (−1)

hεji (m,n,h).

2.4 An application: The algebra End(Pm) as a di-

rect sum of orthogonal SU(2)-subspaces

In this section, following [29, p.580], we express the algebra End(Pm) as a direct sum

of mutually orthogonal SU(2)-subspaces, and find explicit formulae for the projec-

tions on these subspaces. We also give a general method to decompose a matrix

A ∈ End(Pm) into an orthogonal direct sum of matrices, and compute the first two

matrices in this decomposition.

The following proposition is a direct result of Corollary 1.2.16, Lemma 1.2.25,

and Corollary 2.3.8.

Proposition 2.4.1. Let m ∈ N. The algebra End(Pm) can be written uniquely as an

orthogonal direct sum of SU(2)-irreducible subspaces. i.e.

End(Pm) =m⊕t=0

U2t


where U2t : 0 ≤ t ≤ m is a set of SU(2)-irreducible subspaces each of them is iso-

morphic to P2t for some 0 ≤ t ≤ m.

The next proposition gives formulae for the projections of End(Pm) onto the sub-

spaces U2t : 0 ≤ t ≤ m. Recall that by Corollary 2.3.8, the map ηm,m,m−tη∗m,m,m−t

is the projection of Pm⊗Pm onto the SU(2)-invariant subspace that is isomorphic to

P2t.

Proposition 2.4.2. Let m ∈ N, andm⊕t=0

U2t be the decomposition of End(Pm) into a

direct sum of SU(2)-irreducible subspaces. For each 0 ≤ t ≤ m, the map

Vec∗ηm,m,m−tη∗m,m,m−tVec

is the SU(2)-equivariant orthogonal projection of End(Pm) onto U2t.

Proof:

By Lemma 1.2.25, and Lemma 2.3.7, the map

Vec∗ηm,m,m−tη∗m,m,m−tVec : End(Pm) −→ Pm ⊗ Pm −→ End(Pm)

is an SU(2)-equivariant orthogonal projection. Since the map Vec is unitary SU(2)-

equivariant, and the space

Pm ⊗ Pm =m⊕t=0

V2t

with V2t = ηm,m,m−tη∗m,m,m−t(Pm⊗Pm) ' P2t (Corollary 2.3.8), then Vec∗ηm,m,m−tη

∗m,m,m−tVec

is a projection onto a subspace isomorphic to P2t. The result follows by the unique-

ness of the decomposition of End(Pm).

Corollary 2.4.3. Let m ∈ N, and A ∈ End(Pm). The matrix of A can be decomposed

into a direct sum of mutually orthogonal matrices (A0, A2, .....A2m), where

A2t = Vec∗ηm,m,m−tη∗m,m,m−tVec(A).


Namely,

A =m∑t=0

A2t where A2t ∈ U2t, and End(Pm) =m⊕t=0

U2t.

In the rest of this section, we find a general formula to compute the decompo-

sition of a matrix A ∈ End(Pm) into a direct sum of mutually orthogonal matrices

A2t, 0 ≤ t ≤ m . By Corollary 2.4.3, A2t is obtained by applying the two following

steps :

1. Computing the vectors v2t = η∗m,m,m−tVec(A) ∈ P2t for each 0 ≤ t ≤ m.

2. Computing the mutually orthogonal matrices A2t := Av2t = Vec∗ηm,m,m−t(v2t).

For a better explanation of this idea, we find the decomposition for a general

matrix A ∈ End(P1). Recall that ηm,m,m−t = (IPm ⊗ Jm)αm,m,m−t, and that by

Remark 2.1.7, we have Jm(fml ) = (−1)lfmm−l, J∗m(fml ) = (−1)m−lfmm−l

Example 2.4.4. Let A =

a11 a12

a21 a22

∈ End(P1). The matrix A is decomposed

into (A0, A2) where

A0 =

a11+a22

20

0 a11+a22

2

and A2 =

a11−a22

2a12

a21a22−a11

2

Since

〈A0 |A2 〉End(P1) = tr(A∗0A2) = 0

then A0 and A2 are mutually orthogonal.


Proof:

Fix the basis for End(P1) in the following order f 10 f

1∗0 , f

11 f

1∗0 , f

10 f

1∗1 , f

11 f

1∗1 . Let

A =1∑

i,j=0

a(i+1)(j+1)(f1i f

1∗j ). By Corollary 2.4.3, for any t ∈ 0, 1, we have

A2t =1∑

i,j=0

a(i+1)(j+1)Vec∗η1,1,1−tη∗1,1,1−tVec(f 1

i f1∗

j ).

• Case t = 0:

Let εji := εji (1,1,1). Using Corollary 2.3.13, and Remark 2.1.7, we have:

η∗1,1,1Vec(f 1

i f1∗

j ) = η∗1,1,1(f 1

i ⊗ f 1

j ) = α∗1,1,1(IP1 ⊗ J∗1 )(f 1

i ⊗ f 1

j )

= α∗1,1,1(f 1

i ⊗ (−1)1−jf 1

1−j) = (−1)1−jε1−ji−j f

0

i−jδij

= (−1)1−jε1−j1 δijf

0

0

but

Vec∗η1,1,1(f 0

0 ) = Vec∗(IP1 ⊗ J1)α1,1,1(f 0

0 )

= Vec∗(IP1 ⊗ J1)(1∑

s=0

εs0f1

1−s ⊗ f 1

s )

= Vec∗(1∑

s=0

(−1)sεs0f1

1−s ⊗ f 1

1−s)

=1∑

s=0

(−1)sεs0f1

1−sf1∗1−s = ε0

0f1

1 f1∗

1 − ε1

0f1

0 f1∗

0 ;

then,

Vec∗η1,1,1η∗1,1,1Vec(f 1

i f1∗

j ) = (−1)1−jε1−j1 δijVec∗η1,1,1(f 0

0 )

= (−1)1−jε1−j1 δij(ε

0

0f1

1 f1∗

1 − ε1

0f1

0 f1∗

0 )

= (−1)1−jε1−j1 δij(

1√2f 1

1 f1∗

1 + 1√2f 1

0 f1∗

0 ).


Hence,

A0 = Vec∗η1,1,1η∗1,1,1Vec(A) = ( 1√

2f 1

1 f1∗

1 + 1√2f 1

0 f1∗

0 )1∑

i,j=0

a(i+1)(j+1)(−1)1−jε1−j1 δij

= ( 1√2f 1

1 f1∗

1 + 1√2f 1

0 f1∗

0 )(−ε1

0a11 + ε0

0a22)

= ( 1√2f 1

1 f1∗

1 + 1√2f 1

0 f1∗

0 )( 1√2a11 + 1√

2a22)

= a11+a22

2I2.

• Case t = 1 :

Let εji := εji (1,1,0), using Corollary 2.3.13, and Remark 2.1.7, we have

η∗1,1,0Vec(f 1

i f1∗

j ) = η∗1,1,0(f 1

i ⊗ f 1

j ) = α∗1,1,0(IP1 ⊗ J∗1 )(f 1

i ⊗ f 1

j )

= α∗1,1,0(f 1

i ⊗ (−1)1−jf 1

1−j)

= (−1)1−jε1−ji−j+1f

2

i−j+1 · χ[−1,1](i−j+1)

so

η∗1,1,0Vec(f 1

0 f1∗

0 ) = −ε1

1f2

1 , η∗1,1,0Vec(f 1

0 f1∗

1 ) = ε0

0f2

0

and

η∗1,1,0Vec(f 1

1 f1∗

0 ) = −ε1

2f2

2 , η∗1,1,0Vec(f 1

1 f1∗

1 ) = ε0

1f2

1 .

As for 0 ≤ l ≤ 2,

Vec∗η1,1,0(f 2

l ) = Vec∗(IP1 ⊗ J1)α1,1,0(f 2

l ) = Vec∗(IP1 ⊗ J1)(

minl,1∑s=max0,l−1

εsl f1

l−s ⊗ f 1

s )

= Vec∗(


(−1)sεsl f1

l−s ⊗ f 1

1−s) =


(−1)sεsl f1

l−sf1∗

1−s

and

ε00 = 1 = ε1

2, and ε01 = 1√

2= ε1

1,

we have:


1. Vec∗η1,1,0η∗1,1,0Vec(f 1

0 f1∗0 )= (ε1

1)2 f 1

0 f1∗0 − ε0

1ε11f

11 f

1∗1 = 1

2f 1

0 f1∗0 − 1

2f 1

1 f1∗1 .

2. Vec∗η1,1,0η∗1,1,0Vec(f 1

0 f1∗1 )= (ε0

0)2 f 1

0 f1∗1 = f 1

0 f1∗1 .

3. Vec∗η1,1,0η∗1,1,0Vec(f 1

1 f1∗0 )= (ε1

2)2 f 1

1 f1∗0 = f 1

1 f1∗0 .

4. Vec∗η1,1,0η∗1,1,0Vec(f 1

1 f1∗1 )= (ε0

1)2 f 1

1 f1∗1 − ε0

1ε11f

10 f

1∗0 = 1

2f 1

1 f1∗1 − 1

2f 1

0 f1∗0 .

Thus

A2 = a11(12f 1

0 f1∗0 − 1

2f 1

1 f1∗1 ) + a21f

11 f

1∗0 + a12f

10 f

1∗1 + a22

(12f 1

1 f1∗1 − 1

2f 1

0 f1∗0

)=

a11−a22

2a12

a21a22−a11

2

.

The same idea in the example above is used to find the decomposition of any

matrix A in End(Pm) for m ∈ N. By linearity, it is enough to compute the decom-

position of fmi1 fm∗i2

, the basis element for End(Pm). This decomposition is given in

Corollary 2.4.8 below. We begin with the following lemmas which generalize the steps

in the example above.

Lemma 2.4.5. Let m ∈ N, and fmi1 fm∗i2

be an element of the standard basis of

End(Pm). Then

fmi1 fm∗

i2= (ϕt(v2t))0≤t≤m

where

• ϕt : P2t −→ U2t, given by ϕt = V ec∗ηm,m,m−t, and

• v2t = (−1)m−i2εm−i2i1−i2+t(m,m,m−t)f2ti1−i2+t · χ[−t,t](i1−i2)


Proof:

By Corollary 2.4.3,

fmi1 fm∗

i2=(V ec∗ηm,m,m−tη

∗m,m,m−tV ec(f

m

i1fm∗

i2))

0≤t≤m

Let v2t = η∗m,m,m−tVec(fmi1 fm∗i2

) ∈ P2t. Since η∗m,m,m−t = α∗m,m,m−t (IPm ⊗ J∗m), then by

Corollary 2.3.13, and Remark 2.1.7, we have

v2t = (−1)m−i2εm−i2i1−i2+t(m,m,m−t)f2t

i1−i2+t · χ[−t,t](i1−i2)

The result now follows.

Lemma 2.4.6. Let m ∈ N and 0 ≤ t ≤ m. Let ψt : P2t −→ U2t be the map

ψt = V ec∗ηm,m,m−t. For each i such that 0 ≤ i ≤ 2t, we have

ψt(f2t

i ) =

mini+m−t,m∑j=max0,i−t

(−1)jεji (m,m,m−t)fmi+(m−j)−tfm∗

m−j

Proof:

By Remarks 2.3.14 (4),

ηm,m,m−t(f2t

i ) =


(−1)jεji (m,m,m−t)fmi+(m−j)−t ⊗ fmm−j

so

ψt(f2t

i ) = V ec∗ηm,m,m−t(f2t

i ) =


(−1)jεji (m,m,m−t)fmi+(m−j)−tfm∗

m−j

Remark 2.4.7. If in the last lemma i = t, then the matrix ψt(f2tt ) will be a diagonal

matrix given bym∑j=0

(−1)jεjt (m,m,m−t)fmm−jfm∗

m−j


Corollary 2.4.8. Let m ∈ N. For 0 ≤ i1, i2 ≤ m, we have

fmi1 fm∗

i2=(C2t(f

m

i1fm∗

i2))

0≤t≤m

whereC2t(f

mi1fm∗

i2), 0 ≤ t ≤ m

is a set of mutually orthogonal matrices given by

[minm+i1−i2,m∑j=max0,i1−i2

(−1)m−i2+jεm−i2i1−i2+t(m,m,m−t)εj

i1−i2+t(m,m,m−t)fm

i1−i2+(m−j)fm∗

m−j

]· χ[−t,t](i1−i2)

In particular,

fmi fm∗

i =

(m∑j=0

(−1)m−i+jεm−it (m,m,m−t)εjt (m,m,m−t)fmm−jfm∗

m−j

)0≤t≤m

Proof:

For any s, let εsi1−i2+t := εsi1−i2+t(m,m,m−t). By Lemma 2.4.5, and Lemma 2.4.6, we

have

fmi1 fm∗

i2=(C2t(f

mi1fm∗

i2))

0≤t≤m

where C2t(fmi1fm∗

i2)= V ec∗ηm,m,m−t(v2t) is a matrix corresponds to a unique vector

v2t = (−1)m−i2εm−i2i1−i2+tf2t

i1−i2+t · χ[−t,t](i1−i2)

Applying the formula in Remark 2.3.14 (4), we get

C2t(fm

i1fm∗

i2) =

[(−1)m−i2εm−i2i1−i2+tV ec

∗ηm,m,m−t(f2t

i1−i2+t)]· χ[−t,t](i1−i2)

=

[minm+i1−i2,m∑j=max0,i1−i2

(−1)m−i2+jεm−i2i1−i2+tεj

i1−i2+tfm

i1−i2+(m−j)fm∗

m−j

]· χ[−t,t](i1−i2)


Example 2.4.9. Let m ∈ N, and A ∈ End(Pm). If (A2t)0≤t≤m is the decomposition

of A into mutually orthogonal matrices, then

A0 = tr(A)m+1

IPm

Proof:

Let A =m∑

i1,i2=0

a(i1+1)(i2+1)fmi1fm∗

i2. Using Corollary 2.4.3, we have

A0 = Vec∗ηm,m,mη∗m,m,mVec(A) =

m∑i1,i2=0

a(i1+1)(i2+1)Vec∗ηm,m,mη∗m,m,mVec(fmi1 f

m∗

i2)

=m∑

i1,i2=0

a(i1+1)(i2+1)C0(fmi1 fm∗

i2)

By Corollary 2.4.8, for 0 ≤ i1, i2 ≤ m, we have

C0(fmi1 fm∗

i2) =

[m∑j=0

(−1)m−i2+jεm−i20 (m,m,m)εj0(m,m,m)fmm−jfm∗

m−j

]· δi1i2

thus

A0 =m∑i=0

a(i+1)(i+1)

(m∑j=0

(−1)m−i+jεm−i0 (m,m,m)εj0(m,m,m)fmm−jfm∗

m−j

)as

εm−i0 (m,m,m) = (−1)m−i 1√

m+1, and εj0(m,m,m) = (−1)j

1√m+1

, we have

A0 =m∑i=0

a(i+1)(i+1)

(m∑j=0

1m+1

fmm−jfm∗

m−j

)= 1

m+1tr(A)IPm .

Corollary 2.4.10. Let m ∈ N, A ∈ End(Pm), and (A2t)0≤t≤m is the decomposition

of A into mutually orthogonal matrices. If tr(A) 6= 0, then for each 1 ≤ t ≤ m

tr(A2t) = 0


To continue in finding formulae for A2t, 0 < t ≤ m, we apply the same algo-

rithm in the example above. Due to the complicated computations that are needed

to find C2t(fmi1fm∗

i2) for general t, we only compute the matrix A2 (at t = 1) in the

decomposition of A. Before doing so, we need a computational lemma whose proof is

in Appendix B.

Lemma 2.4.11. Let m ∈ Nr0, and εji := εji (m,m,m−1). The following identities hold

1. εj0 = (−1)j

√6(j+1)(m−j)m(m+1)(m+2)

for 0 ≤ j ≤ m− 1.

2. εj1 = (−1)j√

3m(m+1)(m+2)

(m− 2j) for 0 ≤ j ≤ m.

3. εj2 = (−1)j−1

√6j(m−j+1)

m(m+1)(m+2)for 1 ≤ j ≤ m.

The following proposition is a direct result of Corollary 2.4.8, and Lemma 2.4.11.

Note that if fmi1 fm∗i2

/∈ fmi fm∗

i , fmi fm∗i+1, f

mi+1f

m∗i , then |i1 − i2| > 1 and C2(fmi1 f

m∗i2

) will

be a zero matrix.

Proposition 2.4.12. Let m ∈ N. Then

1. C2(fmi fm∗i ) = −3(m−2i)

m(m+1)(m+2)

m∑j=0

(m− 2j)fmm−jfm∗m−j, for 0 ≤ i ≤ m.

2. C2(fmi fm∗i+1) =

6√

(i+1)(m−i)m(m+1)(m+2)

m−1∑j=0

√(j + 1)(m− j)fmm−j−1f

m∗m−j, for 0 ≤ i ≤ m− 1.

3. C2(fmi+1fm∗i ) =

6√

(i+1)(m−i)m(m+1)(m+2)

m∑j=1

√j(m− j + 1)fmm−j+1f

m∗m−j, for 0 ≤ i ≤ m− 1.

4. C2(fmi1 fm∗i2

) is a zero matrix for any fmi1 fm∗i2

/∈ fmi fm∗

i , fmi fm∗i+1, f

mi+1f

m∗i .


Example 2.4.13. Let m ∈ N and, A =m∑

i1,i2=0

a(i1+1)(i2+1)fmi1fm∗

i2∈ End(Pm). Let

(A2t)0≤t≤m be the decomposition of A into mutually orthogonal matrices as in Corol-

lary 2.4.3. Then

A2 = (bkj)1≤k,l≤m+1

where

• bjj = 3(m−2j+2)m(m+2)(m+1)

m+1∑i=1

aii(m− 2i+ 2), for 1 ≤ j ≤ m+ 1.

• bj,j+1 =6√j(m−j+1)

m(m+1)(m+2)

m+1∑i=1

ai,i+1

√i(m− i+ 1), and

bj+1,j =6√j(m−j+1)

m(m+1)(m+2)

m+1∑i=1

ai+1,i

√i(m− i+ 1) for 1 ≤ j ≤ m.

• bkj = 0 elsewhere.

i.e. A2 has the form

b11 b12 0 0 0. . . 0

b12 b22 b23 0 0. . . 0

0 b32 b33 b34 0. . . 0

0 0 b43 b44 b45. . . 0

0 0 0 b54 b55. . . 0

......

......

. . . . . . bm,m+1

0 0 0 0 0 bm+1,m bm+1,m+1

where bjj, bj,j+1, bj+1,j are given above.

Proof:

By Corollary 2.4.3, the matrix

A2 =m∑

i1,i2=0

a(i1+1)(i2+1)Vec∗ηm,m,m−1η∗m,m,m−1Vec(fmi1 f

m∗

i2) =

m∑i1,i2=0

a(i1+1)(i2+1)C2(fmi1 fm∗

i2)


As in Proposition 2.4.12 (4),

C2(fmi1 fm∗i2

) = 0 for fmi1 fm∗i2

/∈ fmi fm∗

i , fmi fm∗i+1, f

mi+1f

m∗i ,

we have

A2 =m∑i=0

ai+1,i+1C2(fmi fm∗

i ) +m−1∑i=0

ai+1,i+2C2(fmi fm∗

i+1) +m−1∑i=0

ai+2,i+1C2(fmi+1fm∗

i )

Using Proposition 2.4.12 (1,2,3), we get:

A2 =m+1∑j=1

(m+1∑i=1

aii3(m−2j+2)(m−2i+2)

m(m+1)(m+2)

)fmj−1f

m∗j−1+

m∑j=1

(m∑i=1

ai,i+16√i(m−i+1)

√j(m−j+1)

m(m+1)(m+2)

)fmj−1f

m∗j

+m∑j=1

(m∑i=1

ai+1,i6√i(m−i+1)

√j(m−j+1)

m(m+1)(m+2)

)fmj f

m∗j−1.

Remark 2.4.14. Sincem+1∑j=1

(m− 2j + 2) = 0, then

tr(A2) =m+1∑j=1

(m+1∑i=1

aii3(m−2j+2)(m−2i+2)

m(m+1)(m+2)

)=

m+1∑i=1

aii(m−2i+2)

m(m+1)(m+2)

m+1∑j=1

(m− 2j + 2) = 0

which is compatible with Corollary 2.4.10.

Chapter 3

Introduction to Quantum Channels

This chapter contains the required definitions and propositions from both operator al-

gebra and the quantum information theory needed to model our examples of quantum

channels. Most of this preliminary information is taken from [16], [27], and [41].

3.1 Positive and completely positive maps

Definition 3.1.1. Let H and K be Hilbert spaces. A linear map Φ : End(H) −→

End(K) is said to be

• positive if Φ(A) ≥ 0 for any positive matrix A ∈ End(H).

• n-positive if Φ⊗ In is positive, where

Φ⊗ In : End(H)⊗Mn (C) −→ End(K)⊗Mn (C)

is the linear map, such that

Φ⊗ In (A⊗B) = Φ(A)⊗B

for A ∈ End(H) and B ∈Mn(C).

• completely positive if it is n-positive for all n ≥ 1.

54

3. Introduction to Quantum Channels 55

It follows from the definition that any completely positive map is positive; how-

ever the converse is not true. The following contains an example of a positive map

that is not completely positive.

Examples 3.1.2.

1. Let H be a Hilbert space. The identity map IEnd(H) : End(H) −→ End(H) is

completely positive.

2. Any ∗−homomorphism is completely positive.

3. The transpose map T : Mn (C) −→Mn (C) defined by taking A 7−→ At,

is an example of a positive map that is not completely positive [28, p.5].

For the proof of the following proposition, see Proposition A.2.3 in Appendix A.

Proposition 3.1.3. Let H and K be Hilbert spaces, and let Φ : End(H) −→ End(K)

be a linear map. If Φ is n-positive, then it is k-positive for all 1 ≤ k ≤ n.

Theorem 3.1.4. [28, p.35]. Let K be a Hilbert space, and n ∈ N. A linear map

Φ : Mn (C) −→ End(K) is completely positive if and only if it is n-positive.

The next theorem, known as the Choi theorem [5, 28, p.35], shows that the

complete positivity of Φ is reflected in its Choi matrix (Lemma 1.2.30).

Theorem 3.1.5. Let H and K be Hilbert spaces. A linear map Φ : End(H) −→

End(K) is completely positive if and only if C(Φ) is a positive element in End(K⊗H).

Definition 3.1.6. Let H and K be Hilbert spaces. A linear map Φ : End(H) −→

End(K) is said to be trace-preserving if tr(Φ(A)) = tr(A) for any A ∈ End(H),

where tr denotes the unnormalized trace on End(H).


3.2 Quantum channels

3.2.1 Quantum systems and quantum states

A quantum system is pair consisting of the arena, where the operations take

place and where the data can be stored, and a state describing this arena known as a

quantum state [27, p.80]. Such a quantum system is represented mathematically by a

complex Hilbert space known as the state space, and a density operator on H known

as the state of H. A formal definition of state is the following:

Definition 3.2.1. Let H be a Hilbert space.

1. A state of H is a density operator % ∈ End(H), i.e, a positive operator in

End(H) that has trace one.

2. A state that is a rank one projection is called a pure state. An impure state is

called a mixed state.

3. The maximally mixed state of H is the state 1dHIH.

The set of all states of H is denoted by D(H), and P(H) denotes the subset of

all pure states of H.

Remark 3.2.2. In operator algebras, the state has a different definition correspond-

ing to the one above, see [10, Ch.6 (6.3 and 6.7)],

Example 3.2.3. The simplest quantum system has a two-dimensional state space.

It is called the qubit, and represented mathematically by C2. The state e1e∗1 =

1 0

0 0

is an example of a pure state of C2. An impure state of C2 is 1

2

1 0

0 1

.

Remarks 3.2.4.


1. The state % ∈ D(H) is pure if and only if % can be written in the form ww∗ for

some unit vector w in H. The fact that w is a unit vector corresponds to the

condition tr(%) = 1 (1 = tr(ww∗) = ‖w‖2).

2. As proved in Section 3.1.2 in [41, p.29], the set of quantum states of a finite-

dimensional Hilbert space H is a compact convex set whose extreme points are

the pure states.

A quantum system that is made up of two or more other quantum systems is

called a composite system. The state space of a composite system is the tensor product

of the state spaces of the components, and the joint state of the total system is the

tensor product of the states of the component [27, p.102]. That is, if %i ∈ D(Hi)

for each 1 ≤ i ≤ m, then %1 ⊗ %2 ⊗ .... ⊗ %m ∈ D(H1 ⊗ H2 ⊗ ... ⊗ Hm). The state

%1 ⊗ %2 ⊗ .... ⊗ %m represents the case where the quantum systems Hi are mutually

independent. If the state of H1 ⊗ H2 ⊗ ... ⊗ Hm can not be expressed as a product

state, then Hi are called correlated.

Definition 3.2.5. Let Hi, 1 ≤ i ≤ m be Hilbert spaces, and %i ∈ D(Hi).

1. The state %1 ⊗ %2 ⊗ .... ⊗ %m is called a product state of the composite system

H1 ⊗H2 ⊗ ...⊗Hm.

2. A separable state is a convex combination of product states of H1⊗H2⊗...⊗Hm.

3. A non-separable state is called an entangled state. A composite system that has

an entangled state is called an entangled system.

Definition 3.2.6. A bipartite quantum system is a composite system that consists

of two quantum systems. A state on bipartite system is called a bipartite state. If

% is a state of H1 ⊗ H2, then %H1 = TrH2(%) and %H2 = TrH1

(%) are states of the

subsystems H1 and H2 respectively. The states %H1 and %H2 are called the reduced

density operators of %.


Example 3.2.7.

The state 12

1 0 0 1

0 0 0 0

0 0 0 0

1 0 0 1

= ww∗ where w = 1√2(e1⊗e1 +e2⊗e2) is an example of

a pure entangled state on the composite system C2 ⊗ C2. This is called a Bell state.

Definition 3.2.8. Let (H, πH) be a representation of a group G. A G-equivariant

state of H is a state of H which is a G-equivariant map.

If D(H)G denotes the G-equivariant states of H, then

D(H)G = End(H)G ∩D(H)

Example 3.2.9. [40] Let H be Hilbert space of dimension n. Let G = U ⊗ U :

U ∈ U(n) where U(n) is the set of all unitary operators on H. A Werner state is

an n× n-dimensional bipartite quantum state that is G-equivariant.

3.2.2 Quantum channels, definition and examples

Sending information from one quantum system to another requires transforma-

tions of the states. In quantum mechanics, a quantum channel (a channel) is defined

to be any method that is used to transfer states between two quantum systems. A

mathematical definition of the quantum channel is the following:

Definition 3.2.10. Let H and K be Hilbert spaces. A quantum channel Φ : End(H) −→

End(K) is a linear completely positive trace-preserving map.

We denote the set of all quantum channels from End(H) toEnd(K) byQC(H,K).

The requirement that Φ is completely positive is justified in the introduction of [19].

Examples 3.2.11. Let H be a Hilbert space.


1. For a unitary operator U on H, the map Φ : End(H) −→ End(H) defined by

Φ(A) = UAU∗ for any A ∈ End(H) is a quantum channel. Such a channel

is called a unitary conjugation channel or briefly a unitary channel. A convex

combination of unitary channels on H is called a random unitary channel. It

is in the form Φ(A) =d∑i=1

piUiAU∗i where pi : 1 ≤ i ≤ d is a probability

distribution, and Ui is a unitary operator on H for each i. The identity map on

End(H) is a special case of the unitary channel.

2. A depolarizing channel is defined for 0 ≤ λ ≤ 1 by

Φλ : End(H) −→ End(H)

A 7−→ λ tr(A)dH

IH + (1− λ)A

where dH is the dimension of H, and IH is the identity map on H.

• For λ = 1, the map Φλ is called the completely depolarizing channel.

• If K is a Hilbert space, the generalized completely depolarizing channel is

the linear map Φ : End(H) −→ End(K) defined by Φ(A) = tr(A)dK

IK for

any A ∈ End(H), where IK is the identity matrix on K, and dk is the

dimension of the space K.

3. For −1dH−1

≤ λ ≤ 1dH+1

, let

Φλ : End(H)→ End(H)

Φλ(A) = λ tr(A)dH

IH + (1− λ)At

where At denotes the transpose matrix of A. The map Φλ is a quantum channel.

This channel is called the transpose depolarizing channel.

4. The partial trace, in Definition 1.2.20, is a quantum channel.


We refer the reader to [12, p.123-129], for the proofs that the examples given

above are channels. The following proposition is straightforward.

Proposition 3.2.12. The tensor product of quantum channels, the composition of

composable quantum channels, and convex linear combinations of quantum channels

are quantum channels.

Proposition 3.2.13. Let H and K be Hilbert spaces. For any α ∈ End(H,K), the

map Adα is a linear completely positive map; it is a channel if and only if α is an

isometry.

Proof:

For A ∈ End(H) such that A ≥ 0, there exist B ∈ End(H) such that A = BB∗.

As

Adα(A) = αAα∗ = αBB∗α∗ = αB (αB)∗ ≥ 0

the map Adα is positive. Let n ∈ N, as End(H)⊗Mn (C) ' End(H ⊗ Cn), and

Adα ⊗ In = Ad(α⊗In)

it follows that Adα ⊗ In is positive. If α is an isometry, then

tr(Adα(A)) = tr(αAα∗) = tr(A)

so Adα is trace preserving.

Definition 3.2.14. [12, p.125].(Unital channel) Let H and K be Hilbert spaces. A

quantum channel Φ : End(H) −→ End(K) is said to be unital if

Φ( 1dHIH) = 1

dKIK

Example 3.2.15. The random unitary channel, and the depolarizing channel are

unital channels.


3.2.3 Characterization of quantum channels

In this section, we give many equivalent representations of the quantum channel.

Definition 3.2.16. Let H and K be Hilbert spaces, and Φ : End(H) −→ End(K)

be a quantum channel. A Stinespring representation (dilation) of Φ is a pair (E,α)

consisting of a Hilbert space E (an environment space), and an isometry α : H −→

K ⊗ E such that Φ(A) = TrE(αAα∗) for any A ∈ End(H). The map TrE denotes

the partial trace over E.

Remark 3.2.17. In operator algebra, a Stinespring representation exists for any

completely positive map, see [28, p.43]. In the case of quantum channels, the partial

trace appears as a consequence of being a trace-preserving map. The author in [16,

p.107-109] explains the link between the two representations in both operator algebra

and quantum information theory.

Definition 3.2.18. Let H and K be Hilbert spaces, and Φ : End(H) −→ End(K) be

a completely positive map. A Kraus representation of Φ is a set of operators

Tj : 1 ≤ j ≤ k ⊂ End(H,K)

that satisfies

Φ(A) =k∑j=1

TjAT∗j

The operators Tj : 1 ≤ j ≤ k are called Kraus operators. If Φ is a quantum channel,

then a Kraus representation Tj : 1 ≤ j ≤ k of Φ is required to satisfy the additional

conditionk∑j=1

T ∗j Tj = IH.

The next proposition gives relationships among the different representations of

a quantum channel and its Choi matrix.


Proposition 3.2.19. Let H and K be Hilbert spaces, and let Φ : End(H) −→

End(K) be a quantum channel.

1. If (E,α) is a Stinespring representation of Φ, and ej : 1 ≤ j ≤ dE is an

orthonormal basis for E, then the maps Tj : H −→ K, 1 ≤ j ≤ dE defined by

Tj = (IK ⊗ e∗j)α

yield a Kraus representation of Φ.

2. If Tj : 1 ≤ j ≤ k is a family of Kraus operators of Φ, then

(a) The space E = Ck, and the map α =k∑j=1

Tj ⊗ ej, where ej : 1 ≤ j ≤ k

is an orthonormal basis element for Ck, form a Stinespring representation

of Φ.

(b) The Choi matrix of Φ is given by C(Φ) =k∑j=1

Vec(Tj)Vec(Tj)∗.

Proof:

Let Tj = (IK ⊗ e∗j)α, and A ∈ End(H). SincedE∑j=1

(IK ⊗ e∗j)B(IK ⊗ ej) =

(IK ⊗ tr) (B) for any B ∈ End(K ⊗ E), then

dE∑j=1

TjAT∗j =

dE∑j=1

(IK ⊗ e∗j)αAα∗(IK ⊗ ej)

= (IK ⊗ tr) (αAα∗) = TrE(αAα∗) = Φ(A)

anddE∑j=1

T ∗j Tj =

dE∑j=1

α∗(I∗K ⊗ ej)(IK ⊗ e∗j)α

= α∗(IK ⊗dE∑j=1

eje∗j)α = α∗(IK ⊗ IE)α = IH.

For the proof of the second statement, see [41, p.51-p.54].

The following theorem summarizes the results in [41, p.51-54].


Theorem 3.2.20. Let H and K be Hilbert spaces, and Φ : End(H) −→ End(K)

be a linear map. The following are equivalent:

1. Φ is a quantum channel.

2. The Choi matrix C(Φ) is a positive element in End(K ⊗ H) that satisfies

TrK(C(Φ)) = IH.

3. Φ has a Stinespring representation (E,α).

4. Φ has a Stinespring representation (E,α) such that dim(E) = rank(C(Φ)).

5. Φ has a Kraus representation.

6. Φ has a Kraus representation T1, T2, ...Tk of Φ where k = rank(C(Φ)).

As a corollary to the equivalence between (1) and (2) in the theorem above, we

have:

Corollary 3.2.21. Let H and K be Hilbert spaces. The set of all quantum channel

Φ : End(H) −→ End(K) can be identified with a proper subset of all the states of

K ⊗ H. Namely, the set % ∈ D(K ⊗ H) : TrK(%) = 1dHIH. This identification is

given by 1dHC, where C is the Choi-Jamiolkowski map.

Remark 3.2.22. For any quantum channel a Stinespring representation is never

unique. In [15], Holevo shows that if (E,α) and (E ′, α′) are two Stinespring represen-

tations of Φ : End(H) −→ End(K), then there exists a partial isometry J : E −→ E ′

such that α′ = (IK ⊗ J)α and α = (IK ⊗ J∗)α′. Stinespring representations with

minimal dimensionality of the space E are called minimal dilations. The next corol-

lary shows that the Stinespring representation with an environment space satisfies

dim(E) = rank(C(Φ)) is a minimal dilation.

By Proposition 3.2.19, and Theorem 3.2.20, we have:


Corollary 3.2.23. Let H and K be Hilbert spaces, and Φ : End(H) −→ End(K)

be a quantum channel. The rank of the Choi matrix of Φ gives an achievable lower

bound for both the number of any Kraus operators of Φ, and of the dimension of any

environment space.

Proof:

Let T1, T2, ....Tk be Kraus operators of Φ. By Proposition 3.2.19, we have:

rank(C(Φ)) = rank

(k∑j=1

vec(Tj)vec(Tj)∗

)≤

k∑j=1

rank (vec(Tj)vec(Tj)∗) = k

By Theorem 3.2.20 (6), this bound is achievable. Let (α,E) be a Stinespring repre-

sentation of Φ. By Proposition 3.2.19 (1), there exist a set of Kraus operators which

has dE elements. Thus, rank(C(Φ)) ≤ dE. By Theorem 3.2.20 (4), this bound is

achievable.

3.3 G-covariant quantum channels

In this section, we restrict our study to a class of quantum channels that are also

G-equivariant maps with respect to a given group G.

Definition 3.3.1. Let (H, πH) and (K, πK) be two unitary representations of a group

G. A quantum channel Φ : End(H) −→ End(K) is G-covariant if

Φ(πH(g)Aπ∗H(g)) = πK(g)Φ(A)π∗K(g)

for each A ∈ End(H) and g ∈ G. If both πH and πK are irreducible representations,

then Φ is called G-irreducibly covariant.

We denote the set of all G-covariant channels from End(H) to End(K) by

QC(H,K)G.


Examples 3.3.2. Let (H, πH) and (K, πK) be two unitary representations of a group

G.

1. The partial trace over H (Definition 1.2.20), is an example of a G-covariant

channel from End(H ⊗ K) to End(K). It is a quantum channel [12, p.124],

with a Stinespring representation (H, IH⊗K), and Kraus operators Tj = IK⊗e∗j :

1 ≤ j ≤ dH, where ej : 1 ≤ j ≤ dH is an orthonormal basis for H. By Lemma

1.2.21, it is a G-equivariant map.

2. The map Φ : End(H⊗K) −→ End(K⊗H) defined by Φ(A) = flipHKA (flipHK)∗

is a unitary conjugation, G-covariant quantum channel. Recall that by Lemma

1.2.23, the map flipHK is a G-equivariant map.

3. The generalized completely depolarizing channel (defined in Example 3.2.11) is

a G-covariant channel.

Proposition 3.3.3. Let G be a group. The tensor product of G-covariant channels,

and the composition of G-covariant channels are again G-covariant channels.

By the convexity of both the set of quantum channels [41, p.49], and the space

of G-equivariant maps, we have :

Proposition 3.3.4. Let (H, πH) and (K, πK) be two representations of a group G.

The set QC(H,K)G of G-covariant channels is a convex set.

As by Example 3.3.2, and Proposition 1.2.7, the partial trace, and the conjugation

map are G-equivariant, the proof of the following proposition is straightforward.

Proposition 3.3.5. Let (H, πH), (K, πK), and (E, πE) be representations of a group

G. Let Φ : End(H) −→ End(K) be a quantum channel given by a Stinespring

representation (E,α). If α : H −→ K ⊗ E is G-equivariant, then Φ is G-covariant.


Recall that if H and K are vector spaces, and W is subspace of H, then the

restriction of a linear map Φ : End(H) −→ End(K) on End(W ), denoted by Φ |W ,

is the map Φ AdιW : End(W ) −→ End(K), where ιW is the inclusion map of W in

H.

Proposition 3.3.6. Let (H, πH) and (K, πK) be representations of a group G. Let

W be a G-invariant subspace of H, and Φ : End(H) −→ End(K) be G-covariant

channel. The restriction of Φ on End(W ) is a G-covariant channel.

Proof:

Let ιW be the inclusion map of W . By Lemma 1.2.10, and Proposition 1.2.7 (3),

the map AdιW : End(W ) −→ End(H) is G-equivariant. By Proposition 3.2.13, it is

a channel. As the composition of two G-covariant channels is G-covariant, the result

follows.

Remark 3.3.7. If (E,α) is a Stinespring representation of Φ, then (E,α ιW ) is a

Stinespring representation of Φ |W . Hence, the proof of the proposition above can be

done using Proposition 3.3.5.

Proposition 3.3.8. Let (H, πH) be a representation of a group G such that H =m⊕i=1

Wi, where Wi are G-invariant subspaces of H. Let qi : 1 ≤ i ≤ m be the

orthogonal projections of H on Wi. The map

Φ : End(H) −→m⊕i=1

End(Wi)

A 7−→m∑i=1

qiAq∗i

is a unital G-covariant channel.


Proof:

By Lemma 1.2.10, qi : 1 ≤ i ≤ m is a set of G-equivariant maps, so by

Proposition 1.2.7 and Proposition 3.2.13, the conjugation map

Adqi : End(H) −→ End(Wi)

A 7−→ qiAq∗i

is a G-equivariant completely positive map; thus so is Φ. As (q∗i qi)mi=1 is a partition

of IH, and (qiq∗i )mi=1 is a partition of I m⊕

i=1Wi

, the map Φ is trace-preserving and unital.

The next proposition is taken from [7, p.6]; we provide a proof for completeness.


group G, and Φ : End(H) −→ End(K) be a linear map. Then

1. Φ is a G-covariant channel if and only if

1

dH

C(Φ) ∈ End(K ⊗H)G ∩% ∈ D(K ⊗H) : TrK(%) =

1

dH

IH

2. If πH is irreducible, then Φ is a G-covariant channel if and only if

1

dHC(Φ) ∈ D(K ⊗H)G.

Proof:

By Corollary 1.2.29, the map 1dHC is a bijective map. Hence

1dHC (A ∩B) = 1

dH(C (A) ∩ C (B))

for any A,B ⊆ End(End(H), End(K)). The first statement follows from this, Propo-

sition 1.2.32, and Corollary 3.2.21. For the second statement, assume πH is an irre-

ducible representation of G. By (1), it is enough to show that

End(K ⊗H)G ∩% ∈ D(K ⊗H) : TrK(%) = 1

dHIH

= D(K ⊗H)G


Since

End(K ⊗H)G ∩% ∈ D(K ⊗H) : TrK(%) = 1

dHIH

⊆ D(K ⊗H)G

always holds, then we only have to prove the other inclusion.

Let % ∈ D(K ⊗H)G. For g ∈ G, we have

% = (πK(g)⊗ πH(g)) % (πK(g)⊗ πH(g))∗

As the partial trace is G-equivariant, we get

TrK(%) = TrK ((πK(g)⊗ πH(g)) % (πK(g)⊗ πH(g))∗) = πH(g)TrK (%) πH(g)∗

So, TrK(%) is intertwining the irreducible representation πH. By Schur’s Lemma

1.2.12, TrK% = λIH for some scalar λ. Taking the trace of both sides, we have

λ = 1dH

. Thus

%∈ End(K ⊗H)G ∩% ∈ D(K ⊗H) : TrK(%) = 1

dHIH

Lemma 3.3.10. Let (H, πH) and (K, πK) be two unitary representations of a group

G, and Φ : End(H) −→ End(K) be a G-covariant channel. If Tj : 1 ≤ j ≤ n are

Kraus operators of Φ, then for each g ∈ G, πK(g)TjπH(g)∗ : 1 ≤ j ≤ n are Kraus

operators of Φ.

Proof:

Let g ∈ G. As πH and πK are unitary representations, then

n∑j=1

(πK(g)TjπH(g)∗)∗ πK(g)TjπH(g)

∗ = πH(g)

(n∑j=1

T ∗j Tj

)πH(g)

∗ = IH

For A ∈ End(H), we have

n∑j=1

πK(g)TjπH(g)∗A (πK(g)TjπH(g)

∗)∗ =n∑j=1

πK(g)TjπH(g)∗AπH(g)T ∗j πK(g)

∗


= πK(g)

(n∑j=1

TjπH(g)∗AπH(g)T ∗j

)πK(g)

∗

= πK(g)Φ(πH(g)∗AπH(g))πK(g)

∗

= πK(g)πK(g)∗Φ(A)πK(g)πK(g)

∗ = Φ(A)

the result follows by Definition 3.2.18.

Chapter 4

EPOSIC Channels

The present chapter introduces EPOSIC channels, examples of SU(2)-covariant

channels. We define these channels in the first section using Stinespring representa-

tion, and study them in the rest of this chapter. In Section 4.2, we obtain Kraus

representation of the EPOSIC channel, and compute its Choi matrix in Section 4.3.

In the next two sections, we compute a complementary channel, and the dual map of

the EPOSIC channel. We end this chapter with an application to operator algebra

by getting an example of a positive map that is not completely positive.


• Constructing EPOSIC channels (Proposition 4.1.1).

• Obtaining a Kraus representation, and the Choi matrix of the EPOSIC channel

(Definition 4.2.1, and Proposition 4.3.5).

• Computing a complementary channel, and the dual map of the EPOSIC channel

(Proposition 4.4.4, and Proposition 4.5.6).

70

4. EPOSIC Channels 71

• Obtaining an example of a positive, non-completely positive map (Proposition

4.6.3, and Proposition 4.6.5).

4.1 EPOSIC channels

Recall the SU(2)-equivariant isometry αm,n,h : Pm+n−2h −→ Pm⊗Pn that was

given in Proposition 2.3.5. According to Definition 3.2.16, and Proposition 3.3.5, this

isometry induces an SU(2)-covariant quantum channel Φm,n,h : End(Pm+n−2h) −→

End(Pm), that has a Stinespring representation (Pn, αm,n,h).

Proposition 4.1.1. For m,n, h ∈ N with 0 ≤ h ≤ minm,n, let r = m + n − 2h

and αm,n,h be as in Definition 2.3.2. The map Φm,n,h : End(Pr) −→ End(Pm) defined

for A ∈ End(Pr) by

Φm,n,h(A) = TrPn(αm,n,hAα∗m,n,h)

is an SU(2)-irreducibly covariant channel.

Definition 4.1.2. Let r,m ∈ N, and

E(r,m) = (n, h) ∈ N2 : r = m+ n− 2h, 0 ≤ h ≤ minm,n

For (n, h) ∈ E(r,m), we call the quantum channel Φm,n,h : End(Pr) −→ End(Pm),

defined in Proposition 4.1.1, an EPOSIC channel.

For r,m ∈ N, we denote by EC(r,m) the set of all EPOSIC channels from

End(Pr) into End(Pm), and abbreviate EC(m,m) to EC(m). As we show in Section

5.1, the set EC(r,m) constitutes the set of extreme points of all SU(2)-irreducibly

covariant channels from End(Pr) into End(Pm), justifying the nomenclature EPOSIC.

Lemma 4.1.3. Let r,m ∈ N. Then

E(r,m) = (r +m− 2l,m− l) ∈ N2 : 0 ≤ l ≤ minr,m


Proof:

Let B = (r +m− 2l,m− l) ∈ N2 : 0 ≤ l ≤ minr,m. Suppose that

(r + m − 2l,m − l) ∈ B for some 0 ≤ l ≤ minr,m. Set n0 = r + m − 2l and

h0 = m − l. The pair (n0, h0) satisfies r = n0 + m − 2h0 and 0 ≤ h0 ≤ minm,n0

(note that h0 ≤ h0 +(r− l)= m− l+r− l = m+r−2l = n0). According to Definition

4.1.2, (r +m− 2l,m− l) = (n0, h0) ∈ E(r,m).

Conversely, assume (n, h) ∈ E(r,m). Set l0 = m − h, then 0 ≤ l0 ≤ m,

n = r−m+2h = r+m−2l0, and l0 ≤ l0 +(n−h) = (m−h)+(n−h)= m+n−2h = r.

Hence, 0 ≤ l0 ≤ minr,m, and (n, h) = (r +m− 2l0,m− l0) ∈ B.

Consequently, we have:

Proposition 4.1.4. Let r,m∈ N. The set of all EPOSIC channels from End(Pr) to

End(Pm) is

EC(r,m) = Φm,r+m−2l,m−l : 0 ≤ l ≤ minr,m

Remark 4.1.5. In [14, p.42], using Schur’s Lemma, it is shown that anyG-irreducibly

covariant channel is unital, see Definition 3.2.14. It follows that the EPOSIC channels

are unital.

4.2 Kraus representations of EPOSIC channels

In this section, we obtain a Kraus representation for the EPOSIC channel Φm,n,h.

We exhibit some of their properties, and show that the obtained Kraus operators has

a symmetric relation.

Definition 4.2.1. Let m,n, h ∈ N with 0 ≤ h ≤ minm,n, and αm,n,h be the

isometry given in Definition 2.3.2. For 0 ≤ j ≤ n, let Tj : Pm+n−2h −→ Pm be the


map defined by

Tj = (IPm ⊗ fn∗

j )αm,n,h

where fnj : 0 ≤ j ≤ n denotes the standard basis for Pn.

In the definition above, we identify the two spaces Pm ⊗ C and Pm through the

unitary map u ⊗ λ 7−→ λu. By Proposition 3.2.19, the set Tj : 0 ≤ j ≤ n is a

Kraus representation of Φm,n,h. We call these Kraus operators, the EPOSIC Kraus

operators.

By direct computations using Corollary 2.3.13 and the definition above, one can

obtain the matrix coefficients of the EPOSIC Kraus operator as follows.

Proposition 4.2.2. Let m,n, h ∈ N with 0 ≤ h ≤ minm,n, and r = m+n−2h. If

Tj : 0 ≤ j ≤ n are the EPOSIC Kraus operators of Φm,n,h, then for each 0 ≤ j ≤ n,

Tj(fri ) =

εjifmlij

if j ∈ B(i)

0 otherwise

where εji and B(i) are given in Notation 2.3.10, and f ks : 0 ≤ s ≤ k is the standard

basis for Pk for k ∈ m, r.

Remarks 4.2.3. With the notation of 2.3.10, we have

j ∈ B(i)⇐⇒ max0, j − h ≤ i ≤ minr,m− h+ j

⇐⇒ max0, h− j ≤ lij ≤ minr − j + h,m

Hence,

1. According to the definition of Tj above, we have

rank(Tj) = dim(Col(Tj)) ≤ |i : j ∈ B(i)| = minm, m+ j − h, r, r − j + h


2. The Kraus operator Tj can be written as

Tj =

minr,m+j−h∑i=max0,j−h

εji fm

lijf r∗i

3. The Kraus operator Tj can also be written as

Tj =

minr−j+h,m∑l=max0,h−j

εjl+j−h fm

l fr∗l+j−h

4. The adjoint map of Tj is given by

T ∗j =


εjl+j−hfr

l+j−hfm∗l

5. Every Tj has a vector representation in Pm ⊗ P r given by

V ec(Tj) =

minr,m−h+j∑i=max0,j−h

εji fm

lij⊗ f ri

Equivalently,

V ec(Tj) =


εjl+j−h fm

l ⊗ f rl+j−h

The next proposition follows directly from Corollary 1.2.19.

Proposition 4.2.4. Let m,n, h ∈ N with 0 ≤ h ≤ minm,n, and r = m + n− 2h.

If Tj : 0 ≤ j ≤ n are the EPOSIC Kraus operators of Φm,n,h, then

〈Tj1 |Tj2 〉End(H,K)= r+1

n+1δj1j2

for each 0 ≤ j1, j2 ≤ n.

Recall by Remark 2.1.4 (1) that for g0 =[

0 1

−1 0

]∈ SU(2), we have

ρm(g0) (fml ) = (−1) lfmm−l


Proposition 4.2.5. Let m,n, h ∈ N with 0 ≤ h ≤ minm,n, r = m + n− 2h, and

Tj : 0 ≤ j ≤ n be the EPOSIC Kraus operators of Φm,n,h. For each 0 ≤ j ≤ n, we

have

ρm(g0)Tjρ∗r(g0) = (−1)j Tn−j

Proof:

Fix j ∈ N, such that 0 ≤ j ≤ n. For each 0 ≤ i ≤ r, we have one of the following

cases:

• If j ∈ B(i), since m− lij = l(r−i)(n−j) then by Corollary 2.3.15 (1), we have:

ρm(g0)Tj(fri ) = (−1)lij εjif

m

m−lij= (−1)lij+h εn−jr−i f

m

l(r−i)(n−j)

= (−1)lij+h Tn−j(fr

r−i) = (−1)jTn−jρr(g0)(f ri )

• If j /∈ B(i), then n− j /∈ n− B(i) = B(r − i) . So, both Tj(fri ) and Tn−j(f

rr−i)

are zero. This implies that the identity also holds in this case.

It is worth noticing that even though Φ is a G-covariant channel, Kraus operators

of Φ are not necessarily G-equivariant. The relation in Proposition 4.2.5 can be

translated into a symmetric relation among the vectors representing the operators Tj

and Tn−j in Pm⊗P r. Recall that by Lemma 1.2.25, the map Vec is SU(2)-equivariant;

thus applying the map Vec on both side of the equation in Proposition 4.2.5, we have:

Vec(Tn−j) = (−1)j (ρm(g0)⊗ ρr(g0)) Vec(Tj) (4.2.1)


The EPOSIC Kraus operators Tj : 0 ≤ j ≤ n, and the maps Jm,Jr (Definition

2.1.5) satisfy

flipPmPr (Jm ⊗ J∗r ) (V ec(Tn−j)) = (−1)m+jV ec(T ∗j )

for each 0 ≤ j ≤ n.


Proof:

By Equation 4.2.1, we have

Vec(Tn−j) = (−1)j (ρm(g0)⊗ ρr(g0)) Vec(Tj)

Thus,

flipPmPr (Jm ⊗ J∗r ) (V ec(Tn−j)) = flipPmPr (Jm ⊗ J∗r ) ((−1)j (ρm(g0)⊗ ρr(g0)) Vec(Tj))

= (−1)jflipPmPr (Jm ⊗ J∗r ) (ρm(g0)⊗ ρr(g0))

(minr−j+h,m∑l=max0,h−j

εjl+j−h fm

l ⊗ f rl+j−h

)

= (−1)j


(−1)j−hεjl+j−hflipPmPr

((Jm ⊗ J∗r )

(fmm−l ⊗ f rr−(l+j−h)

))= (−1)j


(−1)mεjl+j−hflipPmPr

((fml ⊗ f rl+j−h

))= (−1)j+m


εjl+j−h(f rl+j−h ⊗ fml

)= (−1)m+jV ec(T ∗j ).

The following corollary to Proposition 4.2.2 gives the matrix coefficients of the

EPOSIC channel Φm,n,h, with respect to the standard basis of Pm+n−2h.

Corollary 4.2.7. Let m,n, h ∈ N with 0 ≤ h ≤ minm,n, let r = m+ n− 2h. For

0 ≤ i1, i2 ≤ r,

Φm,n,h(fri1f r∗

i2) =

∑j∈B(i1)∩B(i2)

εji1εji2fmli1jf

m∗

li2j

For m ∈ N, and 0 ≤ i ≤ m, the set B(i) associated to the map αm,0,0 is equal to

0. It follows from this and Corollary 4.2.7, that the channel Φm,0,0 is the identity

map on End(Pm). We also have:


Corollary 4.2.8. Let m,n, h ∈ N with 0 ≤ h ≤ minm,n, and r = m+ n− 2h. If

Dk denotes the set of all diagonal operators in End(Pk), then

Φm,n,h(Dr) ⊆ Dm

Corollary 4.2.9. Let m,n, h ∈ N with 0 ≤ h ≤ minm,n, r = m + n − 2h, and

0 ≤ i ≤ r. The sets of non-zero eigenvalues of Φm,n,h(frr−if

r∗r−i), Φm,n,h(f

ri f

r∗i ), and

of Φn,m,h(fri f

r∗i ) coincide. They are all given by

(εji (m,n,h))2 : j ∈ B(i)

.

Proof:

By Corollary 4.2.7, the set (εji (m,n,h))2 : j ∈ B(i) contains all nonzero eigenval-

ues of Φm,n,h(fri f

r∗i ). Let g0 =

(0 1

−1 0

). As

ρr(g0) (f ri ) = (−1)if rr−i

the equivariance of the channel Φm,n,h gives

Φm,n,h(frr−if

r∗

r−i) = Φm,n,h(ρr(g0)f ri fr∗

i (ρr(g0))∗) = ρm(g0)Φm,n,h(fri f

r∗

i )(ρm(g0))∗

So, Φm,n,h(frr−if

r∗r−i) and Φm,n,h(f

ri f

r∗i ) are unitarily conjugate, and have the same

eigenvalues. For the last one, let

• Bm,n,h(i) = j : max0,−m+ i+ h ≤ j ≤ mini + h, n, the set B(i)

associated with αm,n,h, and

• Bn,m,h(i) = l : max0,−n+ i+ h ≤ l ≤ mini + h, m, the set B(i) associ-

ated with αn,m,h.

As l ∈ Bn,m,h(i) if and only if i− l + h ∈ Bm,n,h(i), we have

(εli(n,m,h))2 : l ∈ Bn,m,h(i) = (εli(n,m,h))2 : i−l+h ∈ Bm,n,h(i)

= (εi−j+hi (n,m,h))2 : j ∈ Bm,n,h(i)

= (εliji (n,m,h))2 : j ∈ Bm,n,h(i)

Corollary 2.3.16 (2) finishes the proof.


4.3 The Choi matrix of the EPOSIC channel

In this section, we show that the Choi matrix of Φm,n,h is a multiple of the projection

of Pm ⊗ P r onto the SU(2)-subspace isomorphic to the environment space Pn. The

following proposition is a special case of Proposition 1.2.17, where H = Pm ⊗ P r.

Proposition 4.3.1. Let ρm and ρr be the irreducible representations of SU(2) in Pm

and Pr respectively. Then End(Pm ⊗ P r)SU(2) is an abelian algebra generated by the

projections on the SU(2)-subspaces of Pm ⊗ P r.

Recall by Corollary 2.3.8, that the space Pm⊗P r decomposes into an orthogonal

direct sum of SU(2)-irreducible subspaces Vm+r−2l, 0 ≤ l ≤ minm, r. For each

0 ≤ l ≤ minm, r, the map ηm,r,l : Pm+r−2l −→ Pm ⊗ P r is an isometry whose

image is Vm+r−2l. The maps qm,r,l = ηm,r,lη∗m,r,l where 0 ≤ l ≤ minm, r are the

mutually orthogonal projections onto the subspaces Vm+r−2l. These projections satisfyminm,r∑

l=0

qm,r,l = IPm⊗Pr .

Since Φm,n,h is an SU(2)-covariant channel, by Proposition 4.3.1, and Proposition

1.2.32, we obtain:

Corollary 4.3.2. Let m,n, h ∈ N with 0 ≤ h ≤ minm,n, and r = m + n − 2h.

With the notation above, the Choi matrix of Φm,n,h is a linear combination of qm,r,l :

0 ≤ l ≤ minm, r.

Remark 4.3.3. For m,n, h ∈ N with 0 ≤ h ≤ minm,n, we have:

Pm ⊗ Pm+n−2h 'minm,m+n−2h⊕

l=0

Pn+2(m−h−l)

= Pn+2(m−h) ⊕ Pn+2(m−h−1) ⊕ ......⊕ Pn ⊕ ....⊕ Pn+2(m−h−minm,m+n−2h)

It follows that if r = m + n − 2h, then the set qm,r,l : 0 ≤ l ≤ minm, r contains

qm,r,m−h = ηm,r,m−hη∗m,r,m−h which is the projection on a subspace isomorphic to Pn.

The following lemma will be used in the proof of Proposition 4.3.5.


Lemma 4.3.4. Let m,n, h ∈ N with 0 ≤ h ≤ minm,n, and r = m + n− 2h. The

operator C(Φm,n,h)ηm,r,m−h is nonzero.

Proof:

Let f ks : 0 ≤ s ≤ k be the standard basis for Pk where k ∈ r, n,m. It is

enough to show that C(Φm,n,h) (ηm,r,m−h(fn0 )) 6= 0. By Lemma 1.2.30 and Corollary

4.2.7, we have:

C(Φm,n,h) =r∑

i1,i2=0

Φm,n,h(fr

i1f r∗

i2)⊗ f ri1f

r∗

i2=

r∑i1,i2=0

∑j∈B(i1)∩B(i2)

εji1εji2fmli1jf

m∗

li2j⊗ f ri1f

r∗

i2.

By Lemma 2.3.7 and Corollary 2.3.13, we have:

ηm,r,m−h(fn

0 ) = (IPm ⊗ Jr)αm,r,m−h(fn0 ) = (IPm ⊗ Jr)

(∑t∈B(0)

εt0(m,r,m−h)fml0t ⊗ fr

t

)

=∑t∈B(0)

(−1)tεt0(m,r,m−h)fml0t ⊗ fr

r−t =∑t∈B(0)

λtfm

m−(t+h) ⊗ f rr−t

where λt > 0 (Corollary 2.3.15 (4)).

Hence, we get:

C(Φm,n,h) (ηm,r,m−h(fn

0 )) =r∑

i1,i2=0

∑j∈B(i1)∩B(i2)

εji1εji2fmli1j

fm∗

li2j⊗f ri1f

r∗

i2

(∑t∈B(0)

λtfm

m−(t+h) ⊗ f rr−t

)

=r∑

i1,i2=0

∑j∈B(i1)∩B(i2)

∑t∈B(0)

λtεji1εji2

(fmli1jf

m∗

li2j

)fmm−(t+h)⊗

(f ri1f

r∗

i2

)f rr−t

But for 0 ≤ i1, i2 ≤ r, j ∈ B(i1) ∩B(i2) and t ∈ B(0), we have:

fmli1jfm∗

li2jfmm−(t+h) ⊗ f ri1f

r∗

i2f rr−t =

fmli1n⊗ f ri1 if i2 = r − t, j = n

0 otherwise

.


Since n ∈ B(i1) if and only if n− h ≤ i1 ≤ r, we have

C(Φm,n,h) (ηm,r,m−h(fn

0 )) =r∑

i1=n−h

∑t∈B(0)

λtεn

i1εnr−tf

m

i1−n+h ⊗ f ri1

Assume that

0 = C(Φm,n,h) (ηm,r,m−h(fn

0 )) =r∑

i1=n−h

∑t∈B(0)

λtεn

i1εnr−t f

m

i1−n+h ⊗ f ri1

By the linear independence of fml ⊗ f ri : 0 ≤ l ≤ m, 0 ≤ i ≤ r, we would have∑t∈B(0)

λtεn

i1εnr−t = 0 for n− h ≤ i1 ≤ r

Since by Corollary 2.3.15 (3),

εni1 6= 0 for n− h ≤ i1 ≤ r

we would have∑

t∈B(0)

λtεnr−t = 0.

But by Corollary 2.3.15 (3), we have

εnr−t = (−1)hθt with θt > 0 for t ∈ B(0) = t : 0 ≤ t ≤ m− h.

Therefore, ∑t∈B(0)

λtεn

r−t = (−1)h

m−h∑t=0

λtθt 6= 0

which is a contradiction.



The Choi matrix of the EPOSIC channel Φm,n,h is given by

C(Φm,n,h) =r+1

n+1qm,r,m−h

where qm,r,m−h is the projection of Pm ⊗ P r onto the SU(2)-subspace of dimension

n+ 1.

Proof:

By Corollary 4.3.2, the Choi matrix

C(Φm,n,h) =

minm,r∑l=0

λlqm,r,l

whereqm,r,l = ηm,r,lη

∗m,r,l, 0 ≤ l ≤ minm, r

are the mutually orthogonal projec-

tions onto Vm+r−2l, 0 ≤ l ≤ minm, r. Consequently,

rank(C(Φm,n,h)) =

minm,r∑l=0,λl 6=0

rank(qm,r,l)

By Lemma 4.3.4, we have λm−h 6= 0, hence

rank(C(Φm,n,h)) ≥ rank(qm,r,m−h) = dim(Vn) = n+ 1

Since by Corollary 3.2.23, rank(C(Φm,n,h)) ≤ n+ 1,we have

C(Φm,n,h) = λm−hqm,r,m−h

Taking the trace of both sides of the equation, we get

r + 1 = tr(C(Φm,n,h)) = λm−htr(qm,r,m−h) = λm−hn+ 1

i.e. λm−h = r+1n+1

.


Remark 4.3.6. The above proposition establishes a one-to-one correspondence be-

tween EC(r,m), and the projections on the SU(2)-subspaces of Pm ⊗ P r. This cor-

respondence is given by

Φm,m+r−2l,m−l←→ r+1m+r−2l+1

qm,r,l

Consequently, we have:

Corollary 4.3.7. For r,m ∈ N, there are exactly minr,m+1 elements in EC(r,m).

4.4 A channel complementary to the EPOSIC chan-

nel

Let us first recall the notion of complementary channels given in [15]. Given

three Hilbert spaces H,K and E, and a linear isometry α : H −→ K ⊗ E, the maps

A 7−→ TrE(αAα∗) and A 7−→ TrK(αAα∗) where A ∈ End(H)

define two quantum channels

Φ : End(H) −→ End(K), and Ψ : End(H) −→ End(E)

The maps Φ and Ψ are called mutually complementary. By Theorem 3.2.20,

to any quantum channel, one can associate a complementary channel. However, as

the Stinespring representation (dilation) is not unique, there can be many candidates

for “the” complementary channel. In [15], Holevo shows that if (E,α) and (E ′, α′)

are two Stinespring representations of Φ : End(H) −→ End(K), then there exists a

partial isometry J : E −→ E ′ such that α′ = (IK⊗J)α and α = (IK⊗J∗)α′. It follows

that if ΨE and ΨE′ are complementary channels of Φ, then they are equivalent in the

sense that there exist a partial isometry J : E −→ E ′ such that ΨE(A) = J∗ΨE′(A)J

and ΨE′(A) = JΨE(A)J∗ for any A ∈ End(H). Moreover, if Ψ is a complementary

channel of Φ, then a complementary of Ψ is isometric to Φ [15, p.96].


Definition 4.4.1. Let H and K be Hilbert spaces, Φ : End(H) −→ End(K) be a

quantum channel, and (E,α) be a Stinespring representation of Φ. The map Ψ :

End(H) −→ End(E) sending A 7−→ TrK(αAα∗) is called a complementary channel

of Φ.

It follows from [15, p.96], and [16, p.125].


group G. If Φ : End(H) −→ End(K) is a G-covariant channel, then any channel

complementary to Φ is G-covariant.

The rest of this section is devoted to giving a formula for channel complementary

to EPOSIC channel. The next proposition is needed for the main result of this section.

Proposition 4.4.3. Let m,n, h ∈ N with 0 ≤ h ≤ minm,n. Then

flipPmPn αm,n,h = (−1)hαn,m,h

where αm,n,h is the isometry in Definition 2.3.2.

Proof:

Let r = m + n − 2h, and fks : 0 ≤ s ≤ k be the standard basis of Pk,

k ∈ r,m, n. For 0 ≤ i ≤ r, let

Bm,n,h(i) = j : max0,−m+ i+ h ≤ j ≤ mini+ h, n

and

Bn,m,h(i) = j : max0,−n+ i+ h ≤ j ≤ mini+ h, m

Since j ∈ Bm,n,h(i) if and only if lij = i − j + h ∈ Bn,m,h(i), we have by Corollary

2.3.16 (2),

flipPmPn αm,n,h(fr

i ) =∑

j∈Bm,n,h(i)

εji (m,n,h) fnj ⊗ fmlij = (−1)h

∑lij∈B

n,m,h(i)

εliji (n,m,h) fnj ⊗ fmlij


Let l = lij, then

flipPmPn αm,n,h(fr

i ) = (−1)h∑

l∈Bn,m,h(i)

εli(n,m,h) fni−l+h ⊗ fml

= (−1)h∑

l∈Bn,m,h(i))

εli(n,m,h) fnlil ⊗ fm

l = (−1)hαn,m,h(f

r

i )

Recall that EPOSIC channel Φm,n,h has a Stinespring representation given by

(Pn, αm,n,h) such that αm,n,h : Pm+n−2h −→ Pm ⊗ Pn.

Proposition 4.4.4. Let m,n, h ∈ N with 0 ≤ h ≤ minm,n. The channel Φn,m,h is

a channel complementary to Φm,n,h.

Proof:

By Proposition 4.4.3, and Lemma 1.2.23, we have:

TrPm(αm,n,hAα

∗m,n,h

)= TrPm

(flipPmPn αn,m,hAα

∗n,m,h

(flipPmPn

)∗)= TrPm

(αn,m,hAα

∗n,m,h

)= Φn,m,h(A).

Recall that by Proposition 2.1.6, for any m ∈ N, the map Jm : Pm −→ Pm is unitary.

The following corollary to Proposition 4.4.3 will be used in the next section.

Corollary 4.4.5. For m,n, h ∈ N with 0 ≤ h ≤ minm,n,

flipPmPn (Jm ⊗ J∗n)ηm,n,h = (−1)h ηn,m,h


Proof:


flipPmPn (Jm ⊗ J∗n)ηm,n,h = flipPmPn (Jm ⊗ J∗n)(IPm ⊗ Jn)αm,n,h

= flipPmPn (Jm ⊗ IPn)αm,n,h

By Proposition 2.1.8, and Proposition 4.4.3, we get

flipPmPn (Jm ⊗ J∗n)ηm,n,h = (IPn ⊗ Jm)flipPmPn αm,n,h

= (−1)h(IPn ⊗ Jm)αn,m,h = (−1)

hηn,m,h.

4.5 Duals of EPOSIC channels

In this section, we give a formula for the dual map of Φm,n,h, and investigate when

this map is a channel. We begin by recalling the definition, and some properties of

the dual map of a linear map.

Definition 4.5.1. Let H and K be Hilbert spaces, and Φ : End(H) −→ End(K)

be a linear map. The dual map of Φ is the unique map Φ∗ : End(K) −→ End(H)

satisfying

〈B |Φ(A)〉End(K)

= 〈Φ∗(B) |A〉End(H)

for any A ∈ End(H) and B ∈ End(K), where 〈 · |· 〉End(H)

and 〈 · | ·〉End(K)

are the

Hilbert-Schmidt inner products on End(H) and End(K), respectively.

The proofs of the following propositions are given in Appendix A.


Proposition 4.5.2. Let H and K be Hilbert spaces, and Φ : End(H) −→ End(K)

be a linear map. Then

1. Φ∗ is completely positive if and only if Φ is completely positive.

2. Φ∗ is trace-preserving if and only if Φ(IH) = IK.


group G. A linear map Φ : End(H) −→ End(K) is G-equivariant if and only if Φ∗

is G-equivariant.


be a completely positive map. If Ti : 1 ≤ i ≤ d are Kraus operators of Φ, then

T ∗i : 1 ≤ i ≤ d are Kraus operators of Φ∗.

The rest of this section is devoted to finding the dual map of the EPOSIC channel.

As the Choi-Jamiolkowski map is unitary (Corollary 1.2.29), we obtain a relation

between Φm,n,h and Φ∗m,n,h by examining their Choi matrices, C(Φm,n,h) ∈ End(Pm⊗

P r), and C(Φ∗m,n,h) ∈ End(Pr ⊗ Pm).

Proposition 4.5.5. For m,n, h ∈ N with 0 ≤ h ≤ minm,n, let r = m + n − 2h

and Tm,r =flipPmPr (Jm ⊗ J∗r ). Then

C(Φ∗m,n,h) = Tm,rC(Φm,n,h)T ∗m,r

Proof:

Let Tj : 0 ≤ j ≤ n be the EPOSIC Kraus operators of Φm,n,h, then T ∗j : 0 ≤

j ≤ n are Kraus operators for Φ∗m,n,h. By Propositions 3.2.19 2(b), and Proposition

4.2.6, we have

C(Φ∗m,n,h) =n∑j=0

V ec(T ∗j )(V ec

(T ∗j))∗

=n∑j=0

Tm,rV ec(Tn−j) (Tm,rV ec(Tn−j))∗

=n∑j=0

Tm,rV ec(Tn−j) (V ec(Tn−j))∗ T ∗m,r = Tm,r

(n∑j=0

V ec(Tn−j) (V ec(Tn−j))∗

)T ∗m,r


= Tm,r

(n∑j=0

V ec(Tj) (V ec(Tj))∗

)T ∗m,r = Tm,rC(Φm,n,h)T ∗m,r.

Proposition 4.5.6. Let m,n, h ∈ N with 0 ≤ h ≤ minm,n. Then

Φ∗m,n,h =m+n−2h+1

m+1Φm+n−2h,n,n−h

Proof:

Let r = m+ n− 2h. By Corollary 1.2.29, it suffices to show that

C(Φ∗m,n,h) = r+1m+1

C(Φr,n,n−h)

By Proposition 4.5.5, this is equivalent to show

Tm,rC(Φm,n,h)T ∗m,r = r+1m+1

C(Φr,n,n−h)

By Proposition 4.3.5, it suffices to show that

Tm,rqm,r,m−hT ∗m,r = qr,m,m−h

where qm,r,m−h, and qr,m,m−h are the projections on the SU(2)-irreducible subspaces of

Pm ⊗ P r and Pr ⊗ Pm respectively. By Corollary 2.3.8, and Corollary 4.4.5, one has

Tm,rqm,r,m−hT ∗m,r = Tm,rηm,r,m−hη∗m,r,m−hT ∗m,r

= flipPmPr (Jm ⊗ J∗r ) ηm,r,m−h(flipPmPr (Jm ⊗ J∗r ) ηm,r,m−h

)∗= ηr,m,m−hη

∗r,m,m−h = qr,m,m−h.

By Proposition 4.5.6, we have

Corollary 4.5.7. The dual map of Φm,n,h is a quantum channel if and only if n = 2h.

In this case Φ∗m,2h,h is equal to Φm,2h,h.


4.6 A positive map that is not completely positive

By Proposition 4.1.4, for m ∈ Nr 0, the set of EPOSIC channels from End(P1) to

End(Pm) is given by

EC(1,m) = Φm,m+1,m,Φm,m−1,m−1

Below f 10 , f

11 is the standard basis for P1, given by f 1

0 (x1, x2) = x2 and f 11 (x1, x2) =

x1.

Lemma 4.6.1. Let h ∈ P1 with ‖h‖ = 1. Then

1. There exists gh ∈ SU(2) such that ρ1(gh) (f 10 ) = h.

2. If Φ : End(P1) −→ End(Pm) is an SU(2)-equivariant map, then the matrices

Φ(hh∗) and Φ(f 10 f

1∗0 ) have the same eigenvalues.

Proof:

Assume that h is a unit element in P1; then h can be written as u0f10 + u1f

11 for

some u0, u1 ∈ C that satisfy |u0|2 + |u1|2 = 1. Let gh =

u0 u1

−u1 u0

∈ SU(2). By

Remark 2.1.4 (1), we have

(ρ1(gh)f 1

0 )(x1,x2) = f 1

0 (u0x1 − u1x2, u1x1 + u0x2)

= u1x1 + u0x2 = u0f1

0 (x1,x2) + u1f1

1 (x1,x2)

= h(x1,x2).

For the second statement, by (1), we have

hh∗ = ρ1(gh)f 1

0 f1∗

0 ρ∗1 (gh)

So, by the equivariance property of Φ, we have

Φ(hh∗) = Φ(ρ1(gh)f 1

0 f1∗

0 ρ∗1 (gh)) = ρ1(gh)Φ(f 1

0 f1∗

0 )ρ∗1 (gh)


which gives the result.

By direct computations using the formula for εji in Corollary 2.3.13, and by

Corollary 4.2.7, one can show:

Lemma 4.6.2. Let m ∈ Nr 0. Then

1. Φm,m+1,m(f 10 f

1∗0 ) =

m∑j=0

2(m−j+1)(m+1)(m+2)

fmm−jfm∗m−j.

2. Φm,m−1,m−1(f 10 f

1∗0 ) =

m−1∑j=0

2(j+1)m(m+1)

fmm−j−1fm∗m−j−1.

Proposition 4.6.3. For m ∈ Nr 0 and α ∈ R, the linear map

Φm,m+1,m − αΦm,m−1,m−1 : End(P1) −→ End(Pm)

is positive if and only if α ≤ 1m+2

.

Proof:

Let A be a positive matrix in End(P1). By the spectral theorem, there exist an

orthonormal basis e1, e2 for P1, and non-negative numbers λ1, λ2 such that

A =2∑i=1

λieie∗i .

To show that Φ := Φm,m+1,m−αΦm.m−1,m−1 is positive, it suffices to show the positivity

of Φ(eie∗i ). By Lemma 4.6.1, this is equivalent to the positivity of Φ(f 1

0 f1∗0 ).

By Lemma 4.6.2, we have that

Φ(f 1

0 f1∗

0 ) = Φm,m+1,m(f 1

0 f1∗

0 )− αΦm,m−1,m−1(f 1

0 f1∗

0 ) ≥ 0

if and only if

m∑j=0

2(m−j+1)

(m+1)(m+2)fmm−jf

m∗

m−j − αm−1∑j=0

2(j+1)

m(m+1)fmm−j−1f

m∗

m−j−1 ≥ 0


if and only if

2

m+2fmm f

m∗

m +m−1∑j=0

[2(m−j)

(m+1)(m+2)− α 2(j+1)

m(m+1)

]fmm−j−1f

m∗

m−j−1 ≥ 0.

Thus

Φ(f 1

0 f1∗

0 ) ≥ 0⇐⇒ α ≤ min m(m−j)(m+2) (j+1)

: 0 ≤ j ≤ m− 1.

Since the map f(t) = (m−t)t+1

is decreasing for 0 ≤ t ≤ m− 1, then

min m(m−j)(m+2) (j+1)

: 0 ≤ j ≤ m− 1 = 1m+2

Consequently, Φ(f 10 f

1∗0 ) ≥ 0 if and only if α ≤ 1

m+2.

The following lemma follows by direct computation, using the formula for εji in

Corollary 2.3.13.

Lemma 4.6.4. For m ∈ Nr 0, we have

ε0

1(m,1,0) =√

mm+1

, ε1

1(m,1,0) =√

1m+1

and

ε0

0(m,1,1) =√

1m+1

, ε1

0(m,1,1) = −√

mm+1

Proposition 4.6.5. Let m ∈ Nr 0. For α > 0, the map

Φm,m+1,m − αΦm,m−1,m−1

is not completely positive.

Proof:

Let Φ := Φm,m+1,m− αΦm,m−1,m−1. By Theorem 3.1.5, it is enough to show that

−2αm

is an eigenvalue of C(Φ).

By Proposition 4.3.5,

C(Φm,m+1,m) = 2m+2

ηm,1,0η∗m,1,0


and

C(Φm,m−1,m−1) = 2mηm,1,1η

∗m,1,1

Let v =√m(fm0 ⊗ f 1

0 ) + fm1 ⊗ f 11 , then

C(Φm,m+1,m) (v) = 2m+2

ηm,1,0η∗m,1,0 (

√m(fm0 ⊗ f 1

0 ) + fm1 ⊗ f 11 )

= 2m+2

ηm,1,0 [(−√mε1

1(m,1,0) + ε01(m,1,0))fm+1

1 ]

= 2m+2

ηm,1,0

[(−√

mm+1

+√

mm+1

)fm+11

](Lemma 4.6.4)

= 2m+2

ηm,1,0 [0× fm+11 ]= 0.

Similarly,

C(Φm,m−1,m−1) (v) = 2mηm,1,1η

∗m,1,1 (

√m(fm0 ⊗ f 1

0 ) + fm1 ⊗ f 11 )

= 2m+2

ηm,1,1 [(−√mε1

0(m,1,1) + ε00(m,1,1))fm−1

0 ]

= 2m

( m+1√m+1

)ηm,1,1(fm−10 ) (Lemma 4.6.4)

= 2√m+1m

[1∑j=0

(−1)jεj0fm1−j ⊗ f 1

1−j

]= 2

√m+1m

( 1√m+1

fm1 ⊗ f 11 +

√mm+1

fm0 ⊗ f 10 )

= 2m

(fm1 ⊗ f 11 +√mfm0 ⊗ f 1

0 )= 2mv.

So, we have

C(Φ)(v) = C(Φm,m+1,m)(v)− αC(Φm,m−1,m−1)(v) = −2αmv

Hence, −2αm

is a negative eigenvalue of C(Φ) for any α > 0.

Combining the result of Proposition 3.1.3, and 4.6.3 and Theorem 3.1.4, we

obtain:

Theorem 4.6.6. Let m ∈ Nr 0. For α > 0, the map

Φm,m+1,m − αΦm,m−1,m−1

is not n-positive for any n > 1.

Chapter 5


Channels

Here, we study SU(2)-irreducibly covariant channels. In the first section, we use

results from Section 4.3 to show that the EPOSIC channels EC(r,m) form a spanning

set of the set of all SU(2)-irreducibly equivariant maps End(End(Pr), End(Pm))SU(2).

We also show that they are the extreme points of the set of all SU(2)-irreducibly

covariant channels QC(Pr, Pm)SU(2). In Section 5.2, we decompose SU(2)-irreducibly

covariant channels as orthogonal direct sums of operators between isomorphic SU(2)-

irreducible subspaces of End(Pr) and End(Pm) respectively.


• The EPOSIC channels EC(r,m) constitute the extreme points of the SU(2)-

irreducibly covariant channels QC(Pr, Pm)SU(2) (Corollary 5.1.5).

• Any completely positive SU(2)-irreducibly equivariant map is a multiple of an

SU(2)-covariant channel (Corollary 5.1.6).

92

5. SU(2)-Irreducibly Covariant Channels 93

• Any SU(2)-irreducibly covariant channel is an orthogonal direct sum of opera-

tors (Corollary 5.2.4).

5.1 Extreme points of SU(2)-irreducibly covariant

channels

Recall that for r,m ∈ N, the set EC(r,m) is Φm,r+m−2l,m−l, 0 ≤ l ≤ minr,m , and

QC(Pr, Pm)SU(2) is the set of SU(2)-irreducibly covariant channels from End(Pr) to

End(Pm). In this section, we show that EC(r,m) consist of all the extreme points of

QC(Pr, Pm)SU(2).

Recall also that End(End(Pr), End(Pm))SU(2) denotes the vector space of the

SU(2)-equivariant maps from End(Pr) to End(Pm).

Proposition 5.1.1. Let r,m ∈ N. The set EC(r,m) is a spanning set of

End(End(Pr), End(Pm))SU(2)

.

Proof:

By Proposition 4.3.1, and Proposition 4.3.5, we have

End(Pm ⊗ P r)SU(2) =

minr,m∑

l=0

λlqm,r,l : λl ∈ C

=

minr,m∑

l=0

µlC(Φm,m+r−2l,m−l) : µl ∈ C

= Span C (Φ) : Φ ∈ EC(r,m)

The result now follows from Corollary 1.2.29 and Proposition 1.2.32.


Proposition 5.1.2. Let r,m ∈ N. The set QC(Pr, Pm)SU(2) is the convex hull of

EC(r,m).

Proof:

Let Φ ∈ QC(Pr, Pm)SU(2). Since Φ is an SU(2)-equivariant map, then by Propo-

sition 5.1.1, we have

Φ =

minr,m∑l=0

λlΦm,m+r−2l,m−l λl ∈ C

It remains to show that 0 ≤ λl andminr,m∑

l=0

λl = 1. By Remark 4.3.6, we have

C(Φ) =

minr,m∑l=0

r+1m+r−2l+1

λlqm,r,l

where qm,r,l , 0 ≤ l ≤ minr,m are mutually orthogonal projections of Pm⊗P r. By

the orthogonality of the projections qm,r,l , 0 ≤ l ≤ minr,m, and the positivity of

C(Φ), we have λl ≥ 0 for 0 ≤ l ≤ minm, r.

As Φ and Φm,m+r−2l,m−l are trace preserving, by choosing any state % ∈ D(Pr), we have

1 = tr(%) = tr(Φ(%)) =

minr,m∑l=0

λltr(Φm,m+r−2l,m−l(%)) =

minr,m∑l=0

λl

Since EC(r,m) ⊆ QC(Pr, Pm)SU(2) the result follows.

Proposition 5.1.3. Let r,m ∈ N. Any element in QC(Pr, Pm)SU(2) is uniquely writ-

ten as a convex combination of elements of EC(r,m).

Proof:

By Proposition 5.1.2, we need only to prove the uniqueness. Let Ψ ∈ QC(Pr, Pm)SU(2)

such thatminr,m∑

l=0

λlΦl = Ψ =

minr,m∑l=0

µlΦl


where Φl = Φm,m+r−2l,m−l. By Proposition 4.3.5, and the orthogonality of

qm,r,l, 0 ≤ l ≤ minm, r

we haver+1

m+r−2l+1λlqm,r,l = qm,r,lC(Ψ) =

r+1

m+r−2l+1µlqm,r,l

Thus, λl = µl.

The following definition is a special case of the definition of the extreme sets

given in [31, p.70].

Definition 5.1.4. Let K be a subset of a vector space X, and x ∈ K. we say that

x is an extreme point of K if x is not an internal point of a line interval whose end

points are in K. Analytically, the condition can be expressed as follows:

if y ∈ K, z ∈ K, 0 < t < 1, and x = ty + (1− t)z, then x = y = z.

i.e. x can not be written as a proper convex combination of elements of K other than

itself.

Corollary 5.1.5. Let r,m ∈ N. The set EC(r,m) forms all the extreme points of

QC(Pr, Pm)SU(2).

Proof:

As EC(r,m) ⊂ QC(Pr, Pm)SU(2), by Proposition 5.1.3, we have EC(r,m) are

extreme points of QC(Pr, Pm)SU(2). On the other direction, if Φ is an extreme point

of QC(Pr, Pm)SU(2), then it can not be written as a linear combination of elements of

QC(Pr, Pm)SU(2) other than itself. Hence, Φ must be in EC(r,m).

As EC(r,m) is a spanning set of both End(End(Pr), End(Pm))SU(2), and

QC(Pr, Pm)SU(2), we have the following corollary:


Corollary 5.1.6. Let r,m ∈ N. Any completely positive SU(2)-equivariant linear

map Φ : End(Pr) −→ End(Pm) is a multiple of an SU(2)-irreducibly covariant chan-

nel.

Proof:

By Proposition 5.1.1, we have

Φ =

minr,m∑l=0

λlΦm,r+m−2l,m−l

for some λl ∈ C. Since Φ is completely positive, the coefficients λl are non-negative

(otherwise, C(Φ) =minr,m∑

l=0

r+1m+r−2l+1

λlqm,r,l will have a negative eigenvalue).

Let λ =minr,m∑

l=0

λl.

If λ = 0 then λl = 0 for all 0 ≤ l ≤ minr,m, and Φ = 0 is a multiple of any

SU(2)-irreducibly covariant channel. If λ 6= 0, then

Ψ =

minm,r∑l=0

λlλ

Φm,m+r−2l,m−l

is a convex combination of EPOSIC channels. Thus, by Proposition 5.1.2, Ψ is an

SU(2)-irreducibly covariant channel, and Φ = λΨ.

Remark 5.1.7. For Φ ∈ QC(Pr, Pm)SU(2), the dual map of Φ can be computed using

Proposition 5.1.2, and Proposition 4.5.6. It also follows (by Corollary 4.5.7) that the

map Φ∗ is a quantum channel if and only if Φ ∈ QC(Pm)SU(2).

Recall thatQC(Pm)SU(2), the set of all SU(2)-covariant channels fromEnd(Pm) −→

End(Pm), is the convex hull of EC(m) = Φm,2h,h : 0 ≤ h ≤ m. Let denote the

composition of two channels, then

Proposition 5.1.8. The set(QC(Pm)SU(2),

)is an abelian monoid. It is closed

under involution, in fact every element is self-adjoint.


Proof:

As the composition of two G-covariant channels is again G-covariant, the com-

position of linear maps is an associative binary operation on QC(Pm)SU(2), with the

identity map being the identity element. By Proposition 4.3.1, it is abelian. Since

the dual map of Φm,2h,h is Φm,2h,h (Corollary 4.5.7), then by Proposition 5.1.3 every

element is self adjoint, so QC(Pm)SU(2) is closed under involution.

We complete this section with two examples of SU(2)-irreducibly covariant chan-

nels written as convex combinations of EPOSIC channels.

Example I: The generalized completely depolarizing channel

Recall that the generalized completely depolarizing channel is defined for Hilbert

spaces H and K by

Φ : End(H) −→ End(K)

A −→ tr(A)dK

IK

where IK is the identity matrix on K, and dk is the dimension of the space K. Recall

also that it is a G-covariant channel, see Example 3.3.2. For r,m ∈ N, we denote the

generalized completely depolarizing channel from End(Pr) to End(Pm) by Ψr,m.

Proposition 5.1.9. Let r,m ∈ N. Then

Ψr,m(A) =

minr,m∑l=0

r+m−2l+1(r+1)(m+1)

Φm,r+m−2l,m−l(A).

for any A ∈ End(Pr).


Proof:

By Corollary 5.1.2, there exist λl : 0 ≤ l ≤ minr,m such that

0 ≤ λl ≤ 1,minr,m∑

l=0

λl = 1, and

Ψr,m =

minr,m∑l=0

λl Φm,r+m−2l,m−l

Applying the Choi-Jamiolkowski map on both sides, by Proposition 4.3.5, we get

C(Ψr,m) =

minr,m∑l=0

λlr+1

r+m−2l+1qm,r,l.

On the other hand, by Lemma 1.2.30,

C(Ψr,m) =∑i1,i2

tr(Ei1i2 )

m+1

IPm ⊗ Ei1i2 =∑i1,i2

1

m+1

δi1i2 IPm ⊗ Ei1i2

=∑i1

1

m+1

IPm ⊗ Ei1i1 =1

m+1

IPm ⊗∑i1

Ei1i1 =1

m+1

IPm ⊗ IPr

Thusminr,m∑

l=0

λl(r+1)(m+1)r+m−2l+1

qm,r,l = IPm ⊗ IPr

By the orthogonality of qm,r,l : 0 ≤ l ≤ minr,m, we have

λs(r+1)(m+1)r+m−2s+1

qm,r,s = qm,r,s (IPm ⊗ IPr) = qm,r,s

for 0 ≤ s ≤ minr,m. Taking the trace of both sides, we get λs = r+m−2s+1(r+1)(m+1)

.

Example II: The maps Φ1,m−1,0 Φm,m−1,m−1 and Φ1,m−1,0 Φm,m+1,m

Recall (by Proposition 4.1.4) that for m ∈ Nr 0,

EC(1,m) = Φm,m+1,m,Φm,m−1,m−1, and EC(m, 1) = Φ1,m+1,1,Φ1,m−1,0

Lemma 5.1.10. For m ∈ Nr 0, and 0 ≤ i ≤ m, we have

Φ1,m−1,0(fm

i fm∗

i ) = m−imf 1

0 f1∗0 + i

mf 1

1 f1∗1

where fmi : 0 ≤ i ≤ m is the standard basis for Pm.


Proof:

To simplify the notation, let εji := εji (1,m−1,0) for any j ∈ B(i). For 1 ≤ i ≤ m− 1,

by direct computation, we have

εi−1i =

√im, εii =

√m−im

We also have ε00 = 1, and εm−1

m = 1.

Thus, by Corollary 4.2.7,

Φ1,m−1,0(fm

i fm∗

i ) =∑j∈B(i)

(εji )2fmlijf

m∗

lij=

(ε00)

2f 10 f

1∗0 i = 0

(εi−1i )2f 1

1 f1∗1 + (εii)

2f 10 f

1∗0 1 ≤ i ≤ m− 1

(εm−1m )2f 1

1 f1∗1 i = m

=

f 10 f

1∗0 i = 0

imf 1

1 f1∗1 + m−i

mf 1

0 f1∗0 1 ≤ i ≤ m− 1

f 11 f

1∗1 i = m

= imf 1

1 f1∗1 + m−i

mf 1

0 f1∗

0 .

Proposition 5.1.11. For m ∈ Nr 0, we have

Φ1,m−1,0 Φm,m+1,m = Φ1,2,1


and

Φ1,m−1,0 Φm,m−1,m−1 = m−12m

Φ1,2,1 + m+12m

Φ1,0,0

Proof:

By Proposition 5.1.2, both maps Φ1,m−1,0 Φm,m+1,m, and Φ1,m−1,0 Φm,m−1,m−1

are convex combination of elements in EC(1, 1) = Φ1,2,1,Φ1,0,0. So, there exist

0 ≤ p1, p2 ≤ 1 such that

Φ1,m−1,0 Φm,m+1,m = p1Φ1,2,1 + (1− p1)Φ1,0,0

and

Φ1,m−1,0 Φm,m−1,m−1 = p2Φ1,2,1 + (1− p2)Φ1,0,0

To find p1 and p2, we evaluate the expressions above at f 10 f

1∗0 . By Lemma 4.6.2, and

Lemma 5.1.10, we have

Φ1,m−1,0Φm,m+1,m(f 1

0 f1∗

0 ) = Φ1,m−1,0

(m∑j=0

2(m−j+1)(m+1)(m+2)

fmm−jfm∗

m−j

)

= Φ1,m−1,0

(m∑i=0

2(i+1)(m+1)(m+2)

fmi fm∗

i

)=

m∑i=0

2(i+1)(m+1)(m+2)

Φ1,m−1,0

(fmi f

m∗i

)=

m∑i=0

2(i+1)(m+1)(m+2)

[m−imf 1

0 f1∗0 + i

mf 1

1 f1∗1

]= 2

m(m+1)(m+2)

[m∑i=0

(i+ 1)(m− i)f 1

0 f1∗

0 +m∑i=0

i(i+ 1)f 1

1 f1∗

1

]

= 2m(m+1)(m+2)

[m(m+1)(m+2)

6f 1

0 f1∗

0 + m(m+1)(m+2)3

f 1

1 f1∗

1

]= 1

3f 1

0 f1∗0 + 2

3f 1

1 f1∗

1

On the other hand, direct computations using Corollary 4.2.7, gives

p1Φ1,2,1(f 1

0 f1∗

0 ) + (1− p1)Φ1,0,0(f 1

0 f1∗

0 ) = p1

3f 1

0 f1∗0 + 2p1

3f 1

1 f1∗

1 + (1− p1)f1

0 f1∗

0


Thus p1 = 1. In similar way, by Lemma 4.6.2, we have

Φ1,m−1,0Φm,m−1,m−1(f1

0 f1∗

0 ) = Φ1,m−1,0

(m−1∑j=0

2(j+1)m(m+1)

fmm−j−1fm∗

m−j−1

)

= Φ1,m−1,0

(m−1∑i=0

2(m−i)m(m+1)

fmi fm∗

i

)

=m−1∑i=0

2(m−i)m(m+1)

Φ1,m−1,0

(fmi f

m∗

i

)=

m−1∑i=0

2(m−i)m(m+1)

[(m−i)m

f 1

0 f1∗

0 + imf 1

1 f1∗

1

]= 2

m2(m+1)

[m−1∑i=0

(m− i)2f 1

0 f1∗

0 +m−1∑i=0

i(m− i)f 1

1 f1∗

1

]= 2

m2(m+1)

[m(m+1)(2m+1)

6f 1

0 f1∗

0 + m(m+1)(m−1)6

f 1

1 f1∗

1

]= 2m+1

3mf 1

0 f1∗

0 + m−13m

f 1

1 f1∗

1

On the other hand

p2Φ1,2,1(f 1

0 f1∗

0 ) + (1− p2)Φ1,0,0(f 1

0 f1∗

0 ) = p2

3f 1

0 f1∗0 + 2p2

3f 1

1 f1∗

1 + (1− p2)f1

0 f1∗

0

which implies that p2 = m−12m

.

5.2 SU(2)-irreducibly covariant channel as direct

sum of operators

In this section, we use the decomposition of End(Pr) and End(Pm) in Proposition

2.4.1, to write any SU(2)-irreducibly covariant channel as an orthogonal direct sum

of operators. Recall that if H1, H2, ......Hn is a family of complex Hilbert spaces, then

their orthogonal direct sum, denoted byn⊕i=1

Hi is (h1, h2, ......, hn) : hi ∈ Hi.


Definition 5.2.1. Let H =n⊕i=1

Wi (resp. K =n⊕i=1

Vi) be the orthogonal direct sum of

Hilbert spaces Hi (resp. Ki ) for 1 ≤ i ≤ n, and φi : Wi −→ Vi, 1 ≤ i ≤ n be a

collection of linear maps. The orthogonal direct sum of φi : Wi −→ Vi, 1 ≤ i ≤ n,

denoted byn⊕i=1

φi is the map Φ : H −→ K defined by

Φ((h1, h2, ......, hn)) = (φ1 (h1) , φ2 (h2) , ......, φn (hn))

The proof of the next theorem is provided in the Appendix B.

Theorem 5.2.2. Let G be a group and H =r⊕t=1

Wt (resp. K =m⊕s=1

Vs ), where

Wt : 1 ≤ t ≤ r (resp. Vs : 1 ≤ s ≤ m ) are nonequivalent G-irreducible spaces.

Suppose there exists k ∈ N such that

1. For each 1 ≤ t ≤ k , Wt ' Vt via a G-equivariant isomorphism

ψt : Wt −→ Vt

2. For each t > k the subspace Wt is not equivalent to any of the Vs, for any

1 ≤ s ≤ m.

Then, for any G-equivariant map Φ : H −→ K, there exist λ1, λ2, ..., λk such

that Φ is the orthogonal direct sum of the operators λtψt : 1 ≤ t ≤ k

i.e.,

Φ =k⊕t=1

λtψt

Reminder:

Recall by Proposition 2.4.1 and Proposition 2.4.2, that for m ∈ N, the algebra

End(Pm) decomposes into mutually orthogonal subspaces U2t : 0 ≤ t ≤ m where

U2t = Vec∗mηm,m,m−tη∗m,m,m−tVecm(End(Pm)) = Vec∗mηm,m,m−t(P2t)

and where each matrix A2t in U2t corresponds to a unique vector v2t in P2t.


Lemma 5.2.3. Let r,m ∈ N. Let End(Pr) =r⊕t=0

U2t (resp. End(Pm) =m⊕s=0

U ′2s)

be the decomposition of End(Pr) (resp. End(Pm)) into SU(2)-irreducible invariant

subspaces. For any 0 ≤ t ≤ minm, r, the map

Vec∗mηm,m,m−tη∗r,r,r−tVecr

defines an SU(2)-isomorphism ψ2t : U2t −→ U ′2t. Moreover, both matrices A2t∈ U2t,

and ψ2t(A2t)∈ U ′2t correspond to the same vector in P2t.

Proof:

Let 0 ≤ t ≤ r and 0 ≤ s ≤ m, by Proposition 2.4.2, the SU(2)-irreducible

subspaces U2t and U ′2s are given via the equations U2t = Vec∗rηr,r,r−t(P2t), and U ′2s =

Vec∗mηm,m,m−s(P2s) respectively. For each 0 ≤ t ≤ minm, r, the map

ψ2t = Vec∗mηm,m,m−tη∗r,r,r−tVecr : U2t −→ U ′2t

is a nonzero SU(2)-equivariant map (note that η∗m,m,m−tηm,m,m−t = IP2t= η∗r,r,r−tηr,r,r−t,

so η∗r,r,r−t is onto and ηm,m,m−t 6= 0. Hence, the map ηm,m,m−tη∗r,r,r−t is nonzero). By

Schur’s Lemma 1.2.12, the map ψ2t is isomorphism. To show the second statement,

pick A2t ∈ U2t then v2t = η∗r,r,r−tVecr(A2t) is the vector in P2t corresponding to A2t,

and

ψ2t(A2t) = Vec∗mηm,m,m−tη∗r,r,r−tVecr(A2t) = Vec∗mηm,m,m−t(v2t)

is the matrix in U ′2t ⊆ End(Pm) corresponding to v2t.

Recall by Corollary 2.4.3 that any A ∈ End(Pr) can be decomposed into an

orthogonal direct sum of matrices. Namely A = (A0, A2, ....A2r) where A2t ∈ U2t. By

Theorem 5.2.2 and Lemma 5.2.3, we have

Corollary 5.2.4. Let r,m ∈ N. Let Φ : End(Pr) −→ End(Pm) be an SU(2)-

equivariant map. The map Φ can be written as an orthogonal direct sum of op-

erators. More precisely, there exist λ2t : 0 ≤ t ≤ minm, r, and operators


ψ2t = Vec∗mηm,m,m−tη

∗r,r,r−tVecr : 0 ≤ t ≤ minm, r

such that

Φ (A0 + A2 + ....+ A2r) =

minm,r∑t=0

λ2tψ2t(A2t)

for A0 + A2 + ....+ A2r ∈ End(Pr).

Remark 5.2.5. By Lemma 5.2.3, the matrix ψ2t(A2t) in Corollary 5.2.4 is computed

using the map Vecm∗ηm,m,m−tη

∗r,r,r−tVecr independently from the definition of the map

Φ, and depends only on m,r. i.e. all the information about the map Φ will be encoded

in the set of the numbers λ2t : 0 ≤ t ≤ minr,m.

The rest of this section is devoted to computing the numbers λ2t : 0 ≤ t ≤ minr,m

in Corollary 5.2.4 for the EPOSIC channel Φm,n,h. As Φm,n,h(A2t) = λ2tψ2t(A2t) for

any 0 ≤ t ≤ minr,m, we can find λ2t by evaluating both sides of Φm,n,h(A2t) =

λ2tψ2t(A2t) at some basic element of Pm. The big challenge is making a good choice

for the matrix A2t, the one that minimizes the computations. By Remark 2.4.7, the

matrix A2t that corresponds to the vector f 2tt is a diagonal matrix. Hoping for a good

choice, we choose this matrix to compute λ2t. This matrix is given by the formula

A2t =r∑j=0

(−1)jεjt (r,r,r−t)f rr−jfr∗r−j.

Proposition 5.2.6. Let m,n, h ∈ N with 0 ≤ h ≤ minm,n. Let Φm,n,h be the

associated EPOSIC channel, and r = m + n − 2h. For 0 ≤ t ≤ minm, r, the

coefficient λ2t is given by

λ2t =

r∑j=m−h

(−1)jεjt (r,r,r−t)(εr−j+hr−j (m,n,h)

)2

(−1)mεmt (m,m,m−t)

Proof:

Pick f 2tt ∈ P2t. By Remark 2.4.7, the matrix in End(Pr) that corresponds to f 2t

t

is

A2t =r∑j=0

(−1)jεjt (r,r,r−t)f rr−jfr∗

r−j


By Corollary 4.2.7,

Φm,n,h(fr

r−jfr∗

r−j)(fm

0 ) =

(εr−j+hr−j (m,n,h)

)2fm0 if r−j+h∈B(r−j)

0 else

=

(εr−j+hr−j (m,n,h)

)2fm0 if j≥m−h

0 else

Thus

Φm,n,h(A2t)(fm

0 ) =r∑j=0

(−1)jεjt (r,r,r−t)Φm,n,h(fr

r−jfr∗

r−j)(fm

0 )

=r∑

j=m−h


)2fm0

By Remark 2.4.7, and Lemma 5.2.3, the matrix

ψ2t(A2t) =m∑j=0

(−1)jεjt (m,m,m−t)fmm−jfm∗

m−j

As by Corollary 5.2.4,

Φm,n,h(A2t) = λ2tψ2t(A2t)

we get

λ2t =

r∑j=m−h


)2

(−1)mεmt (m,m,m−t)

For m ∈ N, and 0 ≤ t ≤ m, we have

εmt (m,m,m−t) = (−1)m−tm!√cm,m,m−t

Thus, the formula above for λ2t can simplified:


Corollary 5.2.7. Assuming the same hypothesis as in Proposition 5.2.6, the coeffi-

cient λ2t, is given by

λ2t =

r∑j=m−h

(−1)t+jεjt (r,r,r−t)(εr−j+hr−j (m,n,h)

)2

m!√cm,m,m−t

Next we compute the coefficients λ2t for some EPOSIC channels of small dimen-

sions such as EC(0,m), EC(m, 0), EC(1,m), and EC(m, 1). Before doing so, some

computational lemmas are needed, and can be proved easily using the equations in

Corollary 2.3.16 and the identitym∑k=1

k = m(m+1)2

.

Lemma 5.2.8. Let cm,n,h be as defined in Theorem 2.2.4. For m ∈ N, we have

cm,m,m = 1m!(m+1)!

c0,m,0 = 1(m!)2 cm,m+1,m = 2

m!(m+2)!

cm,m−1,m−1 = 2(m−1)!(m+1)!

c1,m−1,0 = 1m!(m−1)!

c1,m+1,1 = (m+1)m!(m+2)!

cm,m,m−1 = 3(m−1)!(m+2)!

c1,m+1,1 = (m+1)m!(m+2)!

c1,m−1,0 = 1m!(m−1)!

Lemma 5.2.9. Let m ∈ N. then

1. εj0(m,m,m) = (−1)j

√1

m+1, εm−j+1

m−j (1,m+1,1) = −√

m−j+1m+2

for 0 ≤ j ≤ m.

2. If m ≥ 1 , then

(a) εm−jm−j(1,m−1,0) =√

jm

for 1 ≤ j ≤ m.

(b) εj0(m,m−1,m−1) = (−1)j

√2(j+1)m(m+1)

for 0 ≤ j ≤ m− 1.

Example 5.2.10. Let m ∈ N.

1. For the channel Φm,m,m : End(P0) −→ End(Pm) the coefficient is λ0 = 1√m+1

.

2. For the channel Φ0,m,0 : End(Pm) −→ End(P0) the coefficient is λ0 =√m+ 1.


Proof:

1. By Corollary 5.2.7, we have λ0 =ε00(0,0,0)(εm0 (m,m,m))

2

m!√cm,m,m

.

As ε00(0,0,0) = 1, and εm0 (m,m,m) = (−1)m√

m+1, then by Lemma 5.2.8, we get

λ0 =

√m!(m+1)!

(m+1)!= 1√

m+1

2. By Corollary 5.2.7, we have

λ0 =

m∑j=0

(−1)jεj0(m,m,m)

(εm−jm−j(0,m,0)

)2

√c0,0,0

As εj0(m,m,m) = (−1)j√m+1

, εm−jm−j(0,m,0)= 1 and c0,0,0 = 1 then

λ0 =m∑j=0

1√m+1

=√m+ 1


1. For the channel Φm,m+1,m : End(P1) −→ End(Pm) the coefficients are λ0 =√2

m+1and λ2 = −

√2m

3(m+1)(m+2).

2. For the channel Φm,m−1,m−1 : End(P1) −→ End(Pm) the coefficients are λ0 =√2

m+1and λ2 =

√2(m+2)

3m(m+1).

Proof:

For each of the channel Φm,m+1,m and Φm,m−1,m−1, we compute λ2t : t = 0, 1.


1. By Corollary 5.2.7, we have

λ2t =

1∑j=0

(−1)t+jεjt (1,1,1−t)(εm−j+1

1−j (m,m+1,m)

)2

m!√cm,m,m−t

For t = 0, 1. Thus

λ0 =

1∑j=0

(−1)jεj0(1,1,1)

(εm−j+1

1−j (m,m+1,m)

)2

m!√cm,m,m

and

λ2 =

1∑j=0

(−1)j+1εj1(1,1,0)

(εm−j+1

1−j (m,m+1,m)

)2

m!√cm,m,m−1

For 0 ≤ j ≤ 1, we have εm−j+11−j (m,m+1,m) = βm,m+1,m

1−j,m,m−j+1 = (−1)m

√2j!(m−j+1)!

(m+2)!, and

εj0(1,1,1) = (−1)j√2

, εj1(1,1,0) = 1√2, thus

λ0 =

√2

1∑j=0

j!(m−j+1)!

(m+2)!m!√cm,m,m

=√

2 (m+2)m!(m+2)!m!

√cm,m,m

=√

2m+1

and

λ2 =

√2

1∑j=0

(−1)j+1j!(m−j+1)!

m!(m+2)!√cm,m,m−1

=√

2 (−m)m!m!(m+2)!

√cm,m,m−1

= −√

2m3(m+1)(m+2)

2. Similarly, we have

λ2t =

1∑j=1

(−1)t+jεjt (1,1,1−t)(εm−j1−j (m,m−1,m−1)

)2

m!√cm,m,m−t

=(−1)t+1ε1

t (1,1,1−t) (εm−10 (m,m−1,m−1))2

m!√cm,m,m−t

Thus


λ0 =−ε1

0(1,1,1) (εm−10 (m,m−1,m−1))2

m!√cm,m,m

=√

2(m+1)!

√cm,m,m

=√

2m+1

and

λ2 =ε1

1 (1,1,0) (εm−10 (m,m−1,m−1))2

m!√cm,m,m−1

=√

2(m+1)!

√cm,m,m−1

=√

2(m+2)3m(m+1)


1. For the channel Φ1,m+1,1 : End(Pm) −→ End(P1) the coefficients are λ0 =√m+1

2and λ2 = −

√m(m+1)6(m+2)

.

2. For the channel Φ1,m−1,0 : End(Pm) −→ End(P1) the coefficients are λ0 =√m+1

2and λ2 =

√(m+1)(m+2)

6m.

Proof:

1. As

λ2t =

m∑j=0

(−1)t+jεjt (m,m,m−t)(εm−j+1m−j (1,m+1,1)

)2

√c1,1,1−t

we have

λ0 =

m∑j=0

m−j+1

(m+2)√m+1

√c1,1,1

=

√2m∑j=0

m−j+1

(m+2)√m+1

=√

2(m+2)

√m+1

m+1∑j=1

j =√

m+12

and

λ2 =

m∑j=0

(−1)j+1εj1(m,m,m−1)

(εm−j+1m−j (1,m+1,1)

)2

√c1,1,0

=√

2m∑j=0

(−1)j+1(m−j+1)m+2

εj1(m,m,m−1)


= −√

6m∑j=0

(m−j+1)(m−2j)

(m+2)√m(m+1)(m+2)

(see Lemma 2.4.11).

That is

λ2 = − 1(m+2)

√6

m(m+1)(m+2)

m∑j=0

(m− j + 1)(m− 2j)

However

m∑j=0

(m− j + 1)(m− 2j) =m∑j=0

(m2 +m)− (3m+ 2)j + 2j2

= (m2 +m)(m+ 1)− (3m+ 2)m∑j=0

j + 2m∑j=0

j2

= (m2 +m)(m+ 1)− (3m+ 2)m∑j=0

j + 2m∑j=0

j2

= (m2 +m)(m+ 1)− (3m+ 2)m(m+1)2

+ 2m(m+1)(2m+1)6

= m(m+1)(m+2)6

Thus λ2 = −√

m(m+1)6(m+2)

.

2. As

λ2t =

m∑j=1

(−1)t+jεjt (m,m,m−t)(εm−jm−j(1,m−1,0)

)2

√c1,1,1−t

then

λ0 =

m∑j=1

(−1)jεj0(m,m,m)

(εm−jm−j(1,m−1,0)

)2

√c1,1,1

=√

2(m+1)

1m

m∑j=1

j =√

m+12

λ2 =

m∑j=1

(−1)j+1εj1(m,m,m−1)

(εm−jm−j(1,m−1,0)

)2

√c1,1,0


=√

2m

m∑j=1

(−1)j+1jεj1(m,m,m−1)

= −1m

√6

m(m+1)(m+2)

m∑j=1

j(m− 2j)

(see Lemma 2.4.11(2)),

As

m∑j=1

j(m− 2j) = mm∑j=1

j − 2m∑j=1

j2

= m m(m+1)2− 2 m(m+1)(2m+1)

6= −m(m+1)(m+2)

6

we get,

λ2 =

√(m+1)(m+2)

6m

Part II

The Minimal Output Entropy and

The Entanglement Breaking

Property of EPOSIC Channels

112

Chapter 6



Property of Quantum Channels

6.1 The minimal output entropy of quantum chan-

nels

The existence of noise in all information processing systems affects the transmission of

information over a quantum channel. A well-known measure of a channel performance

is the Minimal Output Entropy (MOE). In this section, we give the definition of

minimal output entropy, and exhibit some of its properties.

Recall that if H is a Hilbert space, then D(H) denotes the set of all states of H,

i.e.

D(H) = % ∈ End(H) : % ≥ 0, tr(%) = 1

Definition 6.1.1. Let H be a Hilbert space. The von Neumann entropy is a map

113

6. The Minimal Output Entropy and The Entanglement Breaking Property ofQuantum Channels 114

defined on the state space of H as following

S : D(H) −→ R

S(%) = −tr(% log2%)

where % ∈ D(H).

Remark 6.1.2. For a state %, the von Neumann entropy S(%) =∑i

− λi log2 λi

where λii are the eigenvalues of %. By convention, 0 log2 0 = 0.

Recall that a pure state % of H is a rank one projection in End(H). i.e.

% = ww∗for some unit vector w ∈ H.

Theorem 6.1.3. [27, Thm.11.8, Thm.11.10, Sec.11.3.5]

1. The von Neumann entropy is a concave non-negative function, which is zero if

and only if the state is pure.

2. In a d dimensional Hilbert space H, the von Neumann entropy for a state of H

is at most log2 d. It is log2 d if and only if the state is the maximal mixed state

Idd

.

Lemma 6.1.4. [27, p.514] Let H and K be Hilbert spaces. For σ1⊗σ2 ∈ D(H⊗K),

we have

S(σ1 ⊗ σ2) = S(σ1) + S(σ2)

Definition 6.1.5. (The Minimal Output Entropy) Let H and K be Hilbert spaces,

and Φ : End(H) −→ End(K) be a quantum channel. The minimal output entropy

(MOE), is defined by

Smin(Φ) = minS(Φ(%)) : % is a pure state

where S is the von Neumann entropy.


Remark 6.1.6. In the definition of minimal output entropy [24, 25], the minimum

is taken over all the states in H. However, by the concavity of von Neumann entropy,

the minimal output entropy will be achieved on a pure state.

Notation 6.1.7. Let H and K be Hilbert spaces, and Φ : End(H) −→ End(K) be

a quantum channel with Kraus operators Tj : 1 ≤ j ≤ n. For a pure state % = ww∗

of H, let UΦ,% denote the set uj = Tjw : 1 ≤ j ≤ n.

Remark 6.1.8. With the notation of 6.1.7, for a channel Φ : End(H) −→ End(K),

and a pure state % = ww∗ of H, we have

Φ(ww∗) =n∑j=1

Tjwj (Tjw)∗ =n∑j=1

uju∗j

is a state of K. Hence, the set UΦ,% must contain a nonzero vector.

The following lemma can be proved easily by contradiction.

Lemma 6.1.9. Let H be a Hilbert space, and u, v are non zero vectors of H. If u

and v are linearly independent, then uu∗ and vv∗are linearly independent in End(H).


be a quantum channel. Then

1. Smin(Φ) = 0 if and only if there exist a pure state % of H such that Φ(%) is a

pure state.

2. If for each pure state % of H, the set UΦ,% contains at least two linearly inde-

pendent vectors, then Smin(Φ) 6= 0.

Proof:

By continuity of the von Neumann entropy, and compactness of the set of states

[41, p.29], the minimum entropy is achieved. Thus, if Smin(Φ) = 0 then there is a

state % such that S(Φ(%)) = 0. By Theorem 6.1.3, Φ(%) is a pure state. The other


direction follows from the definition of Smin(Φ). To show the second statement, let %

be a pure state. Since the set UΦ,% has at least two linearly independent vectors, then

by Lemma 6.1.9, Φ(%) =∑

uj∈U%uju

∗j has rank at least two. Hence, Φ(%) is not pure for

any pure state %. The result follows from this and the first statement.

Recall the definition of separable and entangled states, given in Definition 3.2.5.

Proposition 6.1.11. Let H,K and E be Hilbert spaces.

1. If α : H −→ K ⊗ E is an isometry, then the state α%α∗ ∈ D(K ⊗ E) is a pure

state for any pure state % ∈ D(H).

2. Any separable pure state % ∈ D(K ⊗ E) is a product of pure states.

i.e. ∃σ1 ∈ P(K) and σ2 ∈ P(E) such that % = σ1 ⊗ σ2.

Proof:

Let % be a pure state. It straightforward to show that α%α∗ is a state, we also

have (α%α∗)2 = α%α∗α%α∗ = α%α∗, and (α%α∗)∗ = α%α∗. Thus α%α∗ is a projection,

moreover rank(α%α∗) ≤ minrank(α), rank(%), rank(α∗) ≤ rank(%) = 1. Since

there is no state with rank zero, the state α%α∗ must have rank one.

For the second statement, assume that % is a pure state such that % =∑k

λk%k, a

convex combination of product states %k. Since any pure state is an extreme point

in the set of the states [41, p.29], % = %k for each k. That is % = σ1 ⊗ σ2 for some

σ1 ∈ D(K) and σ2 ∈ D(E). By Theorem 6.1.3, and Lemma 6.1.4, we have

0 = S(%) = S(σ1) + S(σ2)

hence S(σ1) = S(σ2) = 0, and both σ1, and σ2 are pure.


Corollary 6.1.12. Let H,K and E be Hilbert spaces, and Φ : End(H) −→ End(K)

be a quantum channel with Stinespring representation (E,α). If Φ has nonzero min-

imal output entropy then for any pure state % ∈ D(H), the state α%α∗ ∈ D(K ⊗ E)

is entangled.

Proof:

Let % be a pure state, and assume to the contrary that α%α∗ is a separable state.

By Proposition 6.1.11, there exist pure states σ1 ∈ P(K), σ2 ∈ P(E) such that

α%α∗ = σ1 ⊗ σ2. Consequently, Φ(%) = TrE(α%α∗) = TrE(σ1 ⊗ σ2) = σ1 is pure. By

Proposition 6.1.10, we get Smin(Φ) = 0.

6.2 The entanglement breaking property of quan-

tum channels

A property of quantum channels that has been studied, and used to classify them is

the elimination of entanglement between the input states of composite systems, see

Definition 3.2.5. Channels having this property are called Entanglement Breaking.

Here is a description by P. Shor [34] of the entanglement breaking channels.

“ Entanglement breaking channels are channels which destroy entangle-

ment with other quantum systems. That is, when the input state is en-

tangled between the input space Hin and another quantum system Href ,

the output of the channel is no longer entangled with the system Href .”

Recall that by Definition 3.2.5, the separable state in a composite system is a

convex combination of product states.

Definition 6.2.1. Let H and K be Hilbert spaces, and Φ : End(H) −→ End(K) be

a completely positive map. We say Φ is entanglement breaking if Φ⊗ In(%) is always


separable for any % ∈ D(H ⊗ Cn), and for any n ∈ N. An entanglement breaking

trace preserving map is abbreviated as E.B.T.

One checks easily (see [20, Thm.3]) that the set of entanglement breaking chan-

nels is a convex set. For a completely positive map Φ, it is evident that the state

Φ⊗ In(%) is separable for any separable state %, so Φ will be entanglement breaking

if Φ ⊗ In maps any entangled state to a separable one. The next proposition is due

to [20, Thm.4], see also [16, Pro.6.22, Pro.6.32].

Proposition 6.2.2. Let H and K be Hilbert spaces. If Φ : End(H) −→ End(K) is

a completely positive map, then the following statements are equivalent

1. Φ is entanglement breaking;

2. Φ⊗ In(uu∗) is separable, where u = 1√dH

dH∑i=1

ei ⊗ ej, and ei : 1 ≤ i ≤ dH is an

orthonormal basis for H;

3. Φ can be written in operator sum form using only Kraus operators of rank one.

Recall the definition of the Choi-Jamiolkowski map, given in 1.2.26. By Lemma

1.2.30, we have Φ⊗ In(uu∗) = 1dHC(Φ), where u = 1√

dH

dH∑i=1

ei ⊗ ei.

Corollary 6.2.3. Let H and K be Hilbert spaces, and Φ : End(H) −→ End(K) be a

completely positive map. The map Φ is entanglement breaking if and only if 1dHC(Φ)

is separable, where C(Φ) is the Choi matrix of Φ.

The proof of the next lemma requires knowledge of the entanglement distillation,

and its relation with separability, see [22], and the introduction in [4]. Assuming this,

the next lemma follows from [21, Thm.1].

Lemma 6.2.4. Let H and K be Hilbert spaces. For % ∈ D(H⊗K), let %H = TrK(%),

and %K = TrH(%) be the reduced density operators of % on the subsystems H and K

respectively. If rank(%) < maxrank(%H), rank(%K), then % is not separable.


The next proposition follows from Lemma 6.2.4, and Corollary 6.2.3. Recall that

if Φ is a quantum channel Φ, then by Theorem 3.2.20, we have TrK(C(Φ)) = IH.

Proposition 6.2.5. Let H and K be Hilbert spaces of dimension dH and dK respec-

tively. Let Φ : End(H)→ End(K) be a quantum channel with Choi matrix C(Φ). If

rank (C(Φ)) < maxdH, rank(TrH(C(Φ))) then Φ is not E.B.T

The following statement, which is generalization of Theorem 6 of [20], is a corol-

lary of both Proposition 6.2.5, and Corollary 3.2.23.

Corollary 6.2.6. Let H and K be Hilbert spaces of dimension dH and dK respectively.

If Φ : End(H) → End(K) is a quantum channel, which can be written using Kraus

operators fewer than maxdH, rank(TrH(C(Φ)) then Φ is not E.B.T.

Recall by Proposition 4.5.4 that if Ti : 1 ≤ i ≤ d are Kraus operators of a

completely positive map Φ, then T ∗i : 1 ≤ i ≤ d are Kraus operators of its dual

map Φ∗.


be a completely positive map. The map Φ is entanglement breaking if and only if its

dual map Φ∗ is entanglement breaking.

Proof:

Assume that Φ is entanglement breaking, so by Proposition 6.2.2, there exist

Kraus operators Tj : 1 ≤ j ≤ k of Φ such that rank(Tj) = 1 for all 1 ≤ j ≤ k.

Since Tj = uv∗ ⇐⇒ T ∗j = vu∗, then By Proposition 4.5.4, the map Φ∗ has Kraus op-

erators T ∗i : 1 ≤ i ≤ d such that rank(T ∗j ) = 1. Hence, it is entanglement breaking.

By exchanging the role of Φ and Φ∗, the result follows.


6.3 The additivity of the classical capacity of quan-

tum channels

In this section, we justify the importance of studying the minimal output en-

tropy, and the E.B.T property for a quantum channel. This section can be skipped

with no impact on the rest of the thesis. The channel’s capacity is defined to be the

maximal rate of reliable communication through a channel [27, p.547]. In [2] Bennett

and Shor classified three distinct capacities of a quantum channel. One of the impor-

tant open questions is that of determining the capability of the channel to transmit

classical information, which is known as the classical capacity of the channel. The

Holevo capacity (product state capacity) is defined to be the classical capacity for

the channel with the restriction that there are no entangled input states are allowed

across many uses of the channel, see [27, p.554]. A fundamental result of quantum

information theory, “The quantum coding theorem” [18], and [32], implies that the

classical capacity of a quantum channel Φ is given by

limn→∞

Cχ(Φ⊗n)

n

where Cχdenotes the Holevo capacity.

In their attempts to increase the capacity of quantum channels, scientists studied

whether running two channels in parallel will increase the total classical capacity of

the two channels. Failing to do so, the capacity is called additive. In general, we have

the following definition:

Definition 6.3.1. Let Q : Φ : Φ is a quantum channel −→ R be a map defined

on the set of quantum channels. Then Q is said to be additive if

Q(Φ1 ⊗ Φ2) = Q(Φ1) +Q(Φ2)

for all quantum channels Φ1 and Φ2.


If the Holevo capacity is additive, then for the independent use for many copies

of a quantum channel Φ, we have Cχ(Φ⊗n) = n · Cχ(Φ). This implies the coincidence

between the classical capacity and Holevo capacity, and consequently implies the

additivity of the classical capacity. The following proposition is an outstanding result

of P. Shor in [36].

Proposition 6.3.2. The Holevo capacity Cχ is additive if and only if the minimal

output entropy Smin is additive.

A conjecture that lasted many years was that the Holevo capacity is additive.

Much research effort was directed to prove this conjecture, and for some special

classes of quantum channels the additivity of minimal output entropy was proved.

For example, tensoring the identity with any channel [1], the unital qubit channel

[24], the depolarizing quantum channel [26], the phase damping channels [26], and

the entanglement breaking channels [34]. However, an outstanding paper in 2008

by Hastings [11] disproved this conjecture. By giving a randomized construction of

channels that violate the additivity of the minimal output entropy, he was able to

show the existence of a channel Φ such that Smin(Φ ⊗ Φ) 6= Smin(Φ) + Smin(Φ).

Since then, the efforts redirected towards constructing an explicit example for the

non-additivity of the minimal output entropy.

Proposition 6.3.3. [17] For an independent use of quantum channels, the minimal

output entropy is sub-additive. That is, if Φ1,Φ2 are two quantum channels that are

used independently, then

Smin(Φ1 ⊗ Φ2) ≤ Smin(Φ1) + Smin(Φ2)

The following proposition summarizes some results in both [15, p.95] and [34].

It presents special cases where the additivity of MOE holds.


Proposition 6.3.4. Let Φ be a quantum channel such that Smin(Φ) = 0 or Φ is an

E.B.T channel. Then, for any arbitrary channel Ψ, we have

Smin(Φ⊗Ψ) = Smin(Ψ⊗ Φ) = Smin(Φ) + Smin(Ψ)

Chapter 7



Property of EPOSIC Channels


• Proving that the minimal output entropy of EPOSIC channel is zero if only and

only if the index h in Φm,n,h is zero (Proposition 7.1.1, and Corollary 7.1.7).

• Computing Smin(Φm,1,1) for m ∈ N (Corollary 7.2.6).

• Finding a lower bound of the minimal output entropy of an element inQC(P1, Pm)SU(2)

(Proposition 7.2.13).

• Examining the E.B.T property of EPOSIC channels (Section 7.3).

123

7. The Minimal Output Entropy and The Entanglement Breaking Property ofEPOSIC Channels 124

7.1 The minimal output entropy of EPOSIC chan-

nel

In this section, we determine the EPOSIC channels with zero minimal output entropy.

Recall that for m ∈ N, the channel Φm,0,0 is the identity channel on End(Pm), and

the channel Φ0,m,0 : End(Pm) −→ End(P0) is given by Φ0,m,0(A) = tr(A). As both

takes a pure state to a pure state, both have a zero minimal output entropy. More

generally, we have the following proposition:

Proposition 7.1.1. For m,n ∈ N, the channel Φm,n,0 has zero minimal output en-

tropy.

Proof:

By Corollary 4.2.7, we have

Φm,n,0(f r0 fr∗

0 ) =0∑j=0

(εj0(m,n,,0))2 fml0jfm∗

l0j= fm0 f

m∗

0

where r = m+n, and fki : 0 ≤ i ≤ k is the standard basis for Pk, k ∈ N. The result

follows by Proposition 6.1.10.

Next, we show that the minimal output entropy of Φm,n,h is not zero if h > 0. The

following lemma can be proved by direct computation using the formula in Corollary

2.3.13.

εji (m,n,h) =


βm,n,hi,s,j

Lemma 7.1.2. For m,n, h ∈ N with 0 ≤ h ≤ minm,n and j ∈ B(i), let εji :=

εji (m,n,h) and r = m+ n− 2h, then we have:

1. ε0i 6= 0, for 0 ≤ i ≤ m− h;

2. εi−m+h

i 6= 0, for m− h ≤ i ≤ r;


3. εi+hi 6= 0, for 0 ≤ i ≤ n− h;

4. εni 6= 0, for n− h ≤ i ≤ r.

Lemma 7.1.3. Let m,n, h ∈ N with 0 < h ≤ minm,n, and r = m + n − 2h. For

any 0 ≤ i ≤ m+ n− 2h, we have

max0,−m+ i+ h < mini+ h, n.

Proof:

Let

j1 = max0,−m+ i+ h =

0 if 0 ≤ i ≤ m− h

i−m+ h if m− h ≤ i ≤ r

and

j2 = mini+ h, n =

i+ h if 0 ≤ i ≤ n− h

n if n− h ≤ i ≤ r

If j1 = 0, then j1 < h ≤ j2. Otherwise,

j1 = i− (m− h) ≤ mini, r −m+ h = mini, n− h < mini+ h, n = j2

Recall the definition of B(i) = j : max0,−m+ i+ h ≤ j ≤ mini + h, n

in Notation 2.3.10.

Corollary 7.1.4. For m,n, h ∈ N with 0 < h ≤ minm,n, let r = m+ n− 2h. For

each 0 ≤ i ≤ r, there exist j1, j2 ∈ B(i) such that j1 < j2, and

εj1i 6= 0, εj2i 6= 0


Proof:

Let j1 = max0,−m+ i+ h and j2 = mini+ h, n. Both j1, j2 ∈ B(i), and by

Lemma 7.1.3 we have j1 < j2. By Lemma 7.1.2, both εj1i 6= 0 and εj2i 6= 0.

Recall by Remark 4.2.3 that EPOSIC Kraus operators of Φm,n,h are given by

Tj =

minr,m+j−h∑i=max0,j−h

εji fm

lijf r∗i

for 0 ≤ j ≤ n. Recall also by Notation 6.1.7, that for a quantum channel Φ with

Kraus operators Tj : 1 ≤ j ≤ n, and for a pure state % = ww∗ of H, the set UΦ,%

denotes the vectors uj = Tjw : 1 ≤ j ≤ n. The proof of the following lemma is via

direct computation.

Lemma 7.1.5. Let m,n, h ∈ N with 0 ≤ h ≤ minm,n, r = m + n − 2h, and

Tj : 0 ≤ j ≤ n be the EPOSIC Kraus operators of Φm,n,h. For w =m+n−2h∑i=0

wifri ∈

Pm+n−2h, we have

Tjw =

minr,m−h+j∑i=max0,j−h

wiεjif

m

i−j+h

Proposition 7.1.6. Let m,n, h ∈ N with 0 < h ≤ minm,n, and Φm,n,h be the

associated EPOSIC channel. For any pure state % ∈ End(Pm+n−2h), the set UΦm,n,h,%

contains at least two linearly independent vectors.

Proof:

Let r = m + n − 2h and % be any pure state in End(Pr). By Remark 3.2.4,

% = ww∗ for some unit vector w ∈ Pr. i.e

w =r∑i=0

wifr

i ,

r∑i=0

|wi|2 = 1

Pick i1 minimal so that wi1 6= 0. By Corollary 7.1.4, there exist j1 < j2 ∈ B(i1) such

that εj1i1 6= 0 and εj2i1 6= 0. Since j ∈ B(i1)⇐⇒ max0, j−h ≤ i1 ≤ minr,m−h+j,


then by Lemma 7.1.5, we have

uj1 = Tj1w =

minr,m−h+j1∑i=max0,j1−h

wiεj1i f

m

i−j1+h 6= 0

and

uj2 = Tj2w =

minr,m−h+j2∑i=max0,j2−h

wiεj2i f

m

i−j2+h 6= 0

If UΦm,n,h,% = uj : 0 ≤ j ≤ n doesn’t contain two linearly independent vectors, then

there exist α 6= 0 such that uj2 = αuj1 . In particular, comparing the coefficients of

fmi1−j2+h, we obtain

0 6= wi1εj2i1

= αwi2εj1i2

for some i2, where i1 − j2 + h = i2 − j1 + h. i.e. i2 = i1 − (j2 − j1) < i1 and wi2 6= 0,

contradicting the minimality of i1.

The following corollary follows directly from the last proposition and Proposition

6.1.10.

Corollary 7.1.7. Let m,n,h∈ N with 0 h ≤ minm,n. Then Smin(Φm,n,h) is

nonzero.

By the last corollary and Corollary 6.1.12, we have:

Corollary 7.1.8. Let m,n,h∈ N with 0 h ≤ minm,n. For any pure state % ∈

D(Pm+n−2h), the state αm,n,h%α∗m,n,h ∈ D(Pm ⊗ Pn) is a pure entangled.

7.2 Examples and special cases

In this section, we give a systematic method to compute the minimal output entropy

of Φm,n,n, for m,n ∈ N, and compute Smin(Φm,1,1). We also give a bound for Smin(Φ),

where Φ : End(P1) −→ End(Pm) is an SU(2)-irreducibly covariant channel.


7.2.1 The minimal output entropy of Φm,1,1

Lemma 7.2.1. Let K be a finite dimensional Hilbert space, and A ∈ End(K) such

that A =n∑j=1

uju∗j . If uj : 1 ≤ j ≤ n are linearly independent vectors in K, then

there exist a basis for K such that the matrix represents A is in the form

〈u1 |u1 〉〈u1 |u2 〉 · · · 〈u1 |un 〉 0 · · · 0

〈u2 |u1 〉〈u2 |u2 〉 · · · 〈u2 |un 〉 0 · · · 0...

......

......

......

......

...

〈un |u1 〉〈un |u2 〉 · · · 〈un |un 〉 0 · · · 0

0 0 0 0 · · · 0...

......

......

0 0 0 0 · · · 0

A matrix in a such form is called Gram matrix.

Proof:

Let un+1, un+2, ....udk be a basis for the orthogonal subspace on spanuj : 1 ≤

j ≤ n and U =u1, u2, ..., un, un+1, ....udk

. The set U forms a basis for K. For each

uk ∈ U , we have

Auk =n∑j=1

uju∗j (uk) =

n∑j=1

〈uj |uk 〉uj if 1 ≤ k ≤ n

0 if k > n

The result follows by writing the matrix for A with respect to the basis U .

Corollary 7.2.2. Let m,n ∈ N with n ≤ m, and % be a pure state of Pm−n. The

matrix representing Φm,n,n(%) is in the form of a Gram matrix.


Proof:

Let Tj : 0 ≤ j ≤ n be the EPOSIC Kraus operators of Φm,n,n : End(Pm−n) −→

End(Pm), and % = ww∗ where w is a unit vector in Pm−n. By Remark 6.1.8,

Φm,n,n(%) = Φm,n,n(ww∗) =n∑j=0

uju∗j

where uj = Tjw . By Lemma 7.1.5,

uj = Tjw =m−n∑i=0

wiεjif

m

i−j+n =

m−j∑k=n−j

wk+j−nεjk+j−nf

m

k

where εji = εji (m,n,n). As the coefficients εji (m,n,n) are nonzero (Corollary 2.3.15 (6)), it

is clear that the set uj : 0 ≤ j ≤ n is linearly independent. The result follows by

Lemma 7.2.1.

Theoretically, we are now able to compute the minimal output entropy for Φm,n,n

by computing the eigenvalues of the Gram matrix of Φm,n,n(%) for any pure state %,

see Definition 6.1.5, and Remark 6.1.2. Due to the complicated computations, we

only find the minimal output entropy of Φm,1,1. Recall that for m ∈ N r 0, the

channel Φm,1,1 : End(Pm−1) −→ End(Pm) has only two EPOSIC Kraus operators

T0, T1, where Tj : Pm−1 −→ Pm, j = 1, 2 as given in Definition 4.2.1. Let % = ww∗

be a pure state in End(Pm−1), and u0 = T0w, u1 = T1w. By Corollary 7.2.2, there is

exist a basis for Pm such that the matrix Φm,1,1(%) is given by

〈u0 |u0 〉〈u0 |u1 〉 0 · · · 0

〈u1 |u0 〉〈u1 |u1 〉 0 · · · 0

0 0 0 · · · 0...

......

...

0 0 0 · · · 0

Using the same notation above, we have


Proposition 7.2.3. Let m ∈ N r 0 and ww∗ be a pure state in End(Pm−1). The

eigenvalues of Φm,1,1(ww∗) are 0, λ1, λ2, where

λ1,2 =1±√

1− 4R

2

with R = ‖u0‖2 ‖u1‖2 − |〈u0 |u1 〉|2.

The proof of the following lemma is purely computational, and is provided in

Appendix B.

Lemma 7.2.4. Let m ∈ N r 0. Using the above notations, the minimal value of

‖u0‖2 ‖u1‖2 − |〈u0 |u1 〉|2 is m(m+1)2 .

Remark 7.2.5. Using Cauchy Schwartz inequality and Lemma 7.2.4, we have

0 ≤ 1−4R ≤ 1 holds for any m ≥ 1 which affirms that λ1,2 are non-negative numbers.

By concavity of von Neumann entropy [27, p.516] and [41, Prop.13.4], we have

that the von Neumann entropy of Φm,1,1(ww∗) achieves its minimum when the dif-

ference between λ1 and λ2 is maximal, this is when R is minimal. Consequently, by

Remark 6.1.2, we get:

Corollary 7.2.6. Let m ∈ Nr 0. Then

Smin(Φm,1,1) = −[1

m+1log2

1

m+1+

m

m+1log2

m

m+1]

with the minimal output entropy achieved at the state % = fm−10 fm−1∗

0 .

7.2.2 Lower bound of the minimal output entropy of an ele-

ment in QC(P1, Pm)SU(2)

Recall that the set QC(P1, Pm)SU(2) consists of all SU(2)-irreducibly covariant chan-

nels Φ : End(P1) −→ End(Pm), and it is the convex hull of the EPOSIC channels


EC(1,m) (Proposition 5.1.2). In this subsection, we find a lower bound of the min-

imal output entropy for these channels assuming m ≥ 5. We begin by recalling

some results from Chapter 4. The following lemma is a reformulation of the result in

Lemma 4.6.1.

Lemma 7.2.7. Let m∈ N and Φ : End(P1) −→ End(Pm) be an SU(2)-equivariant

map. For a unit vector w in P1, the states Φ(ww∗) and Φ(f 10 f

1∗0 ) have the same

eigenvalues.

Corollary 7.2.8. Let m ∈ N and Φ : End(P1) −→ End(Pm) be an SU(2)-covariant

channel. Then

Smin(Φ) = S(Φ(f 1

0 f1∗

0 ))


Example 7.2.9. Let m ∈ Nr 0. Then

1. Smin(Φm,m+1,m) = −m+1∑j=1

2j(m+1)(m+2)

log22j

(m+1)(m+2).

2. Smin(Φm,m−1,m−1) = −m∑j=1

2jm(m+1)

log22j

m(m+1).

Proposition 7.2.10. Let m ∈ Nr0, and Φ : End(P1) −→ End(Pm) be an SU(2)-

covariant channel. There exists 0 ≤ p ≤ 1 such that the eigenvalues of Φ(f 10 f

1∗0 )

are λj = 2(m−j+1)

(m+1)(m+2)p+ 2j

m(m+1)(1− p), 0 ≤ j ≤ m

.

Proof:

Since Φ ∈ QC(P1, Pm)SU(2), then by Proposition 5.1.2, there exists 0 ≤ p ≤ 1

such that

Φ = pΦm,m+1,m + (1− p)Φm,m−1,m−1

By linearity of Φ, and by Lemma 4.6.2, we have

Φ(f 1

0 f1∗

0 ) = pΦm,m+1,m(f 1

0 f1∗

0 ) + (1− p)Φm,m−1,m−1(f 1

0 f1∗

0 )


= 2pm+2

fmm fm∗m +

m∑j=1

[2p(m−j+1)

(m+1)(m+2)+ 2(1−p)j

m(m+1)

]fmm−jf

m∗

m−j

=m∑j=0

[2p(m−j+1)

(m+1)(m+2)+ 2(1−p)j

m(m+1)

]fmm−jf

m∗

m−j

By the above proposition, Corollary 7.2.8, and Remark 6.1.2, we have

Corollary 7.2.11. Let m ∈ N, and Φ : End(P1) −→ End(Pm) be an SU(2)-covariant

channel. There exists 0 ≤ p ≤ 1 such that

Smin(Φ) = −m∑j=0

λj log2 λj

where λj = 2(m−j+1)(m+1)(m+2)

p+ 2jm(m+1)

(1− p).

Remark 7.2.12. The map −x lnx is increasing on [0, 1e], and its integral x2

4− x2 lnx

2

is an increasing map on [0, 1]. Recall that by convention, 0 ln 0 = 0.

Proposition 7.2.13. Let m ∈ N such that m ≥ 5, and Φ : End(P1) −→ End(Pm) be

an SU(2)-covariant channel. Then

Smin(Φ) ≥ 14ln2

(m−2)2

m2(m+1)2

Proof:

By Corollary 7.2.11,

Smin(Φ) = −1ln 2

m∑j=0

λj lnλj

where

λj = 2(m−j+1)(m+1)(m+2)

p+ 2jm(m+1)

(1−p).

Let f(x) = −x lnx, g(x) =∫f(x)dx = x2

4− x2 lnx

2, xj = 2(m−j+1)

(m+1)(m+2), and yj = 2j

m(m+1).

As

xj ≤ 2(m+1)(m+1)(m+2)

= 2m+2

and yj ≤ 2mm(m+1)

= 2m+1

. For m ≥ 5, we have

0 ≤ minxj, yj ≤ pxj + (1− p)yj ≤ maxxj, yj ≤ 2m+1≤ 1

3< 1

e


Thus, by Remark 7.2.12,

f (minxj, yj) ≤ f (pxj + (1− p)yj)

As minxj, yj = yj if and only if j ≤ m2

, and f (pxj + (1− p)yj) ≥ 0 for any j, we

have

ln 2 · Smin(Φ) =m∑j=0

f (pxj + (1− p)yj) ≥bm2 c∑j=0

f (pxj + (1− p)yj)

≥bm2 c∑j=0

f (minxj, yj) =

bm2 c∑j=0

f(

2jm(m+1)

)=

bm2 c∑j=1

f(

2jm(m+1)

)As for x ∈ [0,

⌊m2

⌋], we have 0 ≤ 2x

m(m+1)≤ 1

e. Then for j ∈ [0,

⌊m2

⌋], we have

f(

2jm(m+1)

)=

j∫j−1

f(

2jm(m+1)

)dx ≥

j∫j−1

f(

2xm(m+1)

)dx.

Consequently,

ln 2 · Smin(Φ) ≥bm2 c∑j=1

j∫j−1

f(

2xm(m+1)

)dx =

bm2 c∫0

f(

2xm(m+1)

)dx.

To compute the right hand side, let u = 2xm(m+1)

then

bm2 c∫0

f(

2xm(m+1)

)dx = m(m+1)

2

2bm2 cm(m+1)∫

0

(−u lnu) du = g( 2bm2 cm(m+1)

)− g(0)

but

g( 2bm2 cm(m+1)

) ≥ g( 2(m2 −1)

m(m+1)) = g( m−2

m(m+1)) = ( m−2

m(m+1))2

[14−

ln( m−2m(m+1))

2

]Since m−2

m(m+1)< 1, then g( m−2

m(m+1)) > 1

4

(m−2

m(m+1)

)2

. Thus

Smin(Φ) ≥ 14 ln 2

(m−2

m(m+1)

)2

.


7.3 The entanglement breaking property of EPOSIC

channels

The present section examines the E.B.T property of EPOSIC channel. The E.B.T

property of a quantum channel was reviewed in Section 6.2.

Proposition 7.3.1. For m ∈ N, the channel Φm,m,m and Φ0,m,0 are E.B.T channels.

Proof:

Let Tj : 0 ≤ j ≤ m be the EPOSIC Kraus operator of Φm,m,m. By Remark

4.2.3 (1), we have rank(Tj) ≤ 1 (since r = 0). Thus, Φm,m,m can be written in Kraus

representation using only rank one operators. By Proposition 6.2.2, Φm,m,m is E.B.T.

We also have Φ0,m,0 = (m+ 1)Φ∗m,m,m is an E.B.T channel (see Proposition 6.2.7, and

Proposition 4.5.6).

Since the set of entanglement breaking maps is a convex set [20, Th.3] then by

Proposition 5.1.2, we have:

Corollary 7.3.2. For m ∈ N, let Φ : End(P0) −→ End(Pm) and Ψ : End(Pm) −→

End(P0) be SU(2)-covariant channels. The maps Φ and Ψ are E.B.T channels.

The following proposition determines some special cases where the EPOSIC chan-

nels are not E.B.T. Recall that the EPOSIC channel Φm,n,h has exactly (n + 1)

EPOSIC Kraus operators (Definition 4.2.1).

Proposition 7.3.3. Let m,n, h ∈ N with 0 ≤ h ≤ minm,n. If m > minn, 2h

then Φm,n,h : End(Pm+n−2h) −→ End(Pm) is not E.B.T. In particular, Φm,h,h is not

E.B.T for any 0 ≤ h < m.

Proof:


Assume that m > minn, 2h. If n ≥ 2h, then dim(Pm+n−2h) ≥ dim(Pm). Hence

n+ 1 < m+ 1 ≤ dim(Pm+n−2h) = maxdim(Pm+n−2h), rank(TrPm+n−2h(C(Φm,n,h)))

The result follows by Corollary 6.2.6, and the fact that Φm,n,h has (n + 1) EPOSIC

Kraus operators. If n ≤ 2h, then n ≥ 2(n−h). Since m+n−2h > minn, 2(n−h),

then by Proposition 4.5.6, and the first case, we get that the channel

Φm+n−2h,n,n−h = m+n−2h+1m+1

Φ∗m,n,h

is not E.B.T. The result follows by Proposition 6.2.7.

Chapter 8

The Minimal Output Entropy of

the Tensor Product of


Channels

There are very few tools that can be used to understand the minimal output entropy

of the tensor product of two channels in general. In this chapter, we restrict ourselves

to the SU(2)-irreducibly covariant channels, and obtain a bound on the minimal

output entropy for the tensor product of two of such channels.


• Obtaining a bound on the minimal output entropy for the tensor product of

two SU(2)-irreducibly covariant channels (Corollary 8.2.4).

136

8. The Minimal Output Entropy of the Tensor Product of SU(2)-IrreduciblyCovariant Channels 137

8.1 The tensor product of SU(2)-irreducibly co-

variant channels.

For i = 1, 2, let Hi, Ki be Hilbert spaces, and Φi ∈ End (End(Hi), End(Ki)). Recall

that Φ1⊗Φ2, the tensor product of Φ1 and Φ2, is the endomorphism from End(H1⊗

H2) to End(K1 ⊗K2) such that

Φ1 ⊗ Φ2 (A1 ⊗ A2) = Φ1(A1)⊗ Φ2(A2)

for Ai ∈ End(Hi), i = 1, 2.

Notation 8.1.1. Let H be Hilbert space such that H =m⊕i=1

Wi, and Wi are subspaces

of H. Let P be the set of the corresponding orthogonal projections qi : 1 ≤ i ≤ m.

In this chapter, EP will denote the map EP : End(H) −→m⊕i=1

End(Wi) defined by

EP(A) =m∑i=1

qiAq∗i

For i = 1, 2, and ri,mi ∈ N, let Pri (resp. Pmi) be the SU(2)-irreducible Hilbert

space of dimension ri+1 (resp. mi+1). Let Pr1⊗Pr2 =minr1,r2⊕

k=0

Vk (resp. Pm1⊗Pm2

=

minm1,m2⊕l=0

Wl ) be the decomposition of Pr1⊗Pr2 (resp. Pm1⊗Pm2

) into a direct sum of

S(2)-irreducible subspaces. By Proposition 1.2.5, Proposition 3.3.6, and Proposition

3.3.8, we have

Proposition 8.1.2. With the notation above, if Φi : End(Pri) −→ End(Pmi), i = 1, 2

are SU(2)-covariant channels, then

1. For 0 ≤ k ≤ minr1, r2, the restriction of Φ1 ⊗ Φ2 to End(Vk)

Φ1 ⊗ Φ2 |End(Vk) : End(Vk) −→ End(Pm1⊗ Pm2

)

is an SU(2)-covariant channel.


2. If P = ql : 0 ≤ l ≤ minm1,m2 is the set of orthogonal projections of

Pm1⊗ Pm2

on the SU(2)-irreducible subspaces Wl, then the map

EP (Φ1 ⊗ Φ2 |End(Vk)) : End(Vk) −→minm1,m2⊕

l=0

End(Wl)

is an SU(2)-covariant channel.

Proposition 8.1.3. Let EP (Φ1 ⊗ Φ2 |End(Vk)) be the channel given in Proposition

8.1.2. For each 0 ≤ k ≤ minr1, r2, we have:

EP (Φ1 ⊗ Φ2 |End(Vk)) =

minm1,m2∑l=0

λk,lψk,l

where for 0 ≤ l ≤ minm1,m2, the map ψk,l : End(Vk) −→ End(Wl) is an SU(2)-

irreducibly covariant channel, and λk,l : 0 ≤ l ≤ minm1,m2 are non-negative real

numbers withmin(m1,m2∑

l=0

λk,l = 1.

Proof:

The map

ql (Φ1 ⊗ Φ2 |End(Vk)) q∗l : End(Vk) −→ End(Wl)

is a completely positive SU(2)-irreducibly equivariant map (note that this map-

ping is not necessarily trace preserving). By Corollary 5.1.6, it is a multiple of

an SU(2)-irreducibly covariant channel, i.e. there exist SU(2)-covariant channel

ψk,l : End(Vk) −→ End(Wl), and a non-negative number λk,l such that

ql (Φ1 ⊗ Φ2 |End(Vk)) q∗l = λk,lψk,l

Consequently,

EP (Φ1 ⊗ Φ2 |End(Vk)) =

minm1,m2∑l=0

ql (Φ1 ⊗ Φ2 |End(Vk)) q∗l

=

minm1,m2∑l=0

λk,lψk,l


By taking the trace of both sides for any state %, we get that

minm1,m2∑l=0

λk,l = 1

8.2 Bound for the minimal output entropy of the

tensor product of two SU(2)-irreducibly covari-

ant channels.

Recall the definition of von Neumann entropy, and the minimal output entropy given

in Section 6.1. Recall also if W is a subspace of a Hilbert space H, then End(W )

isomorphic to a subspace of End(H) via the map A 7−→

A 0

0 0

, see also Remark

1.2.11.

Lemma 8.2.1. Let H and K be Hilbert spaces, and W be a subspace of H. Let

Φ : End(H) −→ End(K) be a quantum channel. Then

Smin(Φ) ≤ Smin(Φ |End(W ))

Proof:

Let % ∈ End(W ) be a pure state, % can be considered as a pure state in End(H).

Thus,

Smin(Φ) = minS(%) : % ∈ P(H) ≤ minS(%) : % ∈ P(W ) = Smin(Φ |End(W ))


The following result is proved in [12, p.226].

Lemma 8.2.2. Let H and K be Hilbert spaces, and Φ : End(H) −→ End(K) be a

unital quantum channel. Then S(%) ≤ S(Φ(%)) for % ∈ D(H), where S is the von

Neumann entropy.

Proposition 8.2.3. Let H,K and L be Hilbert spaces. Let Φ : End(H) −→ End(K)

be a quantum channel, and Ψ : End(K) −→ End(L) be a unital quantum channel.

Then S(Φ(%)) ≤ S(Ψ(Φ(%))) for % ∈ D(H). Consequently

Smin(Φ) ≤ Smin(Ψ Φ).

Recall by Corollary 2.3.6 that for ri,mi ∈ N, i = 1, 2, the space Pr1 ⊗ Pr2 =minr1,r2⊕

k=0

Vk (resp. Pm1⊗ Pm2

=minm1,m2⊕

l=0

Wl ), where Vk (resp. Wl) is an SU(2)-

irreducible space isomorphic to Pr1+r2−2k(resp Pm1+m2−2l). Keeping these notations,

we have

Corollary 8.2.4. Let Φ1 : End(Pr1) −→ End(Pm1) and Φ2 : End(Pr2) −→ End(Pm2

)

be SU(2)-covariant channels. For each 0 ≤ k ≤ minr1, r2, we have

Smin(Φ1 ⊗ Φ2) ≤ Smin

(minm1,m2∑

l=0

λk,lψk,l

)

where ψk,l : End(Vk) −→ End(Wl), 0 ≤ l ≤ minm1,m2 are SU(2)-irreducibly

covariant channels, and λk,l are non-negative real numbers withmin(m1,m2∑

l=0

λk,l = 1.

Proof:

Pick k such that 0 ≤ k ≤ minr1, r2, let Vk be the SU(2)-irreducible subspace

of Pr1 ⊗ Pr2 of dimension r1 + r2 − 2k. By Lemma 8.2.1,

Smin(Φ1 ⊗ Φ2) ≤ Smin(Φ1 ⊗ Φ2 |End(Vk))


Let P = ql : 0 ≤ l ≤ minm1,m2 be the SU(2)-equivariant projections of Pm1⊗

Pm2onto the subspaces Wl. By Proposition 3.3.8, the map EP : End(Pm1

⊗Pm2) −→

m⊕i=1

End(Wi) defined by

EP(A) =minm1,m2∑

l=1

qlAq∗l

is a unital channel, so by Proposition 8.2.3, and Proposition 8.1.3, we have

Smin(Φ1 ⊗ Φ2) ≤ Smin(EP(Φ1 ⊗ Φ2 |End(Vk))) = Smin

(minm1,m2∑

l=0

λk,lψk,l

)

Part III

Appendices

142

Appendix A

Background Results in Operator

Algebras

A.1 Background definitions and lemmas

As in the rest of the thesis, we assume all vector spaces to be complex vector spaces

of finite dimension.

Definition A.1.1. [31, p.4] Let H be a vector space. A norm on H is a map

‖·‖ : H −→ [0,∞)

such that

1. ‖x+ y‖ ≤ ‖x‖+ ‖y‖ for all vectors x, y in H.

2. ‖αx‖ = |α| ‖x‖ if x in H and α is a scalar.

3. ‖x‖ > 0 if x 6= 0 where x in H.

A vector space that equipped with a norm is called a normed space.

Definition A.1.2. Let H be a vector space. An inner (scalar) product on H is a

map 〈· |·〉 : H ×H −→ C, which is

143

A. Background Results in Operator Algebras 144

1. linear in the second argument,

i.e. 〈x |λy 〉 = λ 〈x |y 〉 and 〈x |y + z 〉 = 〈x |y 〉 + 〈x |z 〉, for all x, y, z in H and

all scalars λ ∈ C.

2. conjugate symmetric,

i.e. 〈x |y 〉 = 〈y |x〉, for all x, y in H.

3. positive definite, i.e. 〈x |x〉 ≥ 0 and 〈x |x〉 = 0 if and only if x = 0 for all x in

H.

A vector space H with an inner product is called an inner product space.

Remark A.1.3. In the definition above, we follow the definition that most physicists

use, as given in [10, p.1].

Lemma A.1.4. Let H be a vector space. An inner product on H induces a norm on

H, given by ‖x‖ =√〈x |x〉 for x ∈ H.

Definition A.1.5. A Hilbert space is an inner product space that is complete as a

normed space (with respect to the norm induced by its inner product).

Definition A.1.6. Let H be a vector space. The conjugate space H is the vector

space with the same underlying abelian group as H, and with scalar multiplication

(λ, v) 7−→ λ.v = λv.

If H is a Hilbert space, then H is also a Hilbert space endowed with the inner

product defined by 〈h1 |h2 〉H = 〈h2 |h1 〉H.

Notation A.1.7. For a Hilbert space H, let H∗ denote the vector space of all linear

forms on H. The space H∗ is called the dual space of H.

Theorem A.1.8. [6, p.40] Let H be a Hilbert space. The map

T : H −→ H∗


h 7−→ fh : H −→ C

x 7−→ 〈h |x〉H

is isometric anti-isomorphism from H to H∗.

For vector spaces H and K, let End(H,K) denote the vector space of linear

maps from H to K. We write End(H) for End(H,H) and IH for the identity map

on H.

Definition A.1.9. Let H be a Hilbert space, and e1, e2, ...en be an orthonormal

basis of H. The un-normalized trace on End(H), is the linear map tr : End(H) −→ C

defined by

tr(T ) =n∑i=1

〈ei |Tei 〉H

The definition of the trace does not depend on the choice of the basis.

Lemma A.1.10. Let H be a Hilbert space. For T ∈ End(H), we have

tr(T ∗) = tr(T ) = tr(T ).

For Hilbert spaces H and K, the space End(H,K) endowed with the Hilbert-

Schmidt inner product given by 〈A |B 〉End(H,K)

= tr(A∗B) for A,B ∈ End(H,K) is a

Hilbert space.

Remark A.1.11. For T ∈ End(H,K), u ∈ H and λ ∈ C, we have

T (λ.u) = T (λu) = λT (u) = λ.T (u)

Hence, the map T is also linear regarded as a map from H to K. We use T to denote

T when it is regarded as a map of H to K.

Proposition A.1.12. Let H and K be Hilbert spaces. The Hilbert spaces End(H,K)

and End(H,K) are equal, and the Hilbert Schmidt inner product on them is the same.


Definition A.1.13. For Hilbert spaces H, K and T ∈ End(H,K), the adjoint of T

is the unique linear map T ∗ ∈ End(K,H) such that

〈y |Tx〉K

= 〈T ∗y |x〉H

∀x ∈ H, ∀y ∈ K.

Definition A.1.14. [10, p.6-8]Let H be a Hilbert space, and T ∈ End(H). The

operatorT is called Hermitian (self adjoint) if T = T ∗. A positive operator T (denoted

by T ≥ 0), is a Hermitian operator with non-negative eigenvalues. An orthogonal

projection T is a positive operator with an eigenvalues belongs to 0, 1 . i.e. T 2 =

T = T ∗. Finally, T is a unitary operator if T satisfy T T ∗ = T ∗T = IH.

Lemma A.1.15. [10, p.6] Let H be a Hilbert space, and T ∈ End(H). The following

are equivalent:

1. T is a positive operator.

2. T = SS∗ for some S ∈ End(H).

3. 〈Tw |w 〉H≥ 0 for any w ∈ H.

Lemma A.1.16. Let K be a Hilbert space. For T ∈ End(K) and 0 6= w ∈ K, we

have

〈T |ww∗ 〉End(K)

= 〈Tw |w 〉K.

Proof:

Let fi : 1 ≤ i ≤ dK be an orthonormal basis for K with f1 = w‖w‖ . We hence

have:

〈T |ww∗ 〉End(K)

= tr(T ∗ww∗) =dK∑i=1

〈fi |T ∗ww∗fi 〉K

= 〈f1 |T ∗ww∗f1 〉K =⟨

w‖w‖

∣∣∣T ∗ww∗ ( w‖w‖

)⟩K

= 〈w |T ∗w 〉K

= 〈Tw |w 〉K.


Proposition A.1.17. Let K be a Hilbert space, and T ∈ End(K). The operator T

is positive if and only if 〈T |X 〉End(K)

≥ 0 for all positive X.

Proof:

If T and X are positive, then T = T1T∗1 , X = X1X

∗1 for some T1, X1 ∈ End(K),

and

〈T |X 〉End(K)

= 〈T1T∗1 |X1X

∗1 〉End(K)

= tr (T1T∗1X1X

∗1 ) = tr ((T ∗1X1)∗ T ∗1X1) ≥0.

For the other direction, assume that 〈T |X 〉End(K)

≥ 0 for all positive X. As ww∗ is

positive for any w ∈ K, we have

〈Tw |w 〉K

= 〈T |ww∗ 〉End(K)

≥ 0

and T is positive.

Definition A.1.18. Let H be a Hilbert space, and W1, W2 be subspaces of H. We

say the spaces W1 and W2 are mutually orthogonal if 〈u1 | u2〉H = 0 for every u1 ∈ W1

and u2 ∈ W2.

Recall that if H1, H2, ....., Hn Hilbert spaces, their (external) direct sum denoted

byn⊕i=1

Hi, is the (h1, h2, ....., hn) : hi ∈ Hi. It is a Hilbert space with the inner

product given by 〈(h1, h2, ....., hn) | (k1, k2, ....., kn)〉 =n∑i=1

〈hi | ki〉Hi .

Definition A.1.19. Let H be a Hilbert space, and H1, H2, ....., Hn be subspaces of H,

such that H = H1 +H2 + · · · · ·+Hn, and Hi ∩

(∑i 6=jHj

)= 0, then H is called the

(internal) direct sum of Hi : 1 ≤ i ≤ n. If H1, H2, ....., Hn are mutually orthogonal

subspaces of H, then H is called the orthogonal direct sum of Hi : 1 ≤ i ≤ n.

If H1, H2, ....., Hn are subspaces of a Hilbert space H, satisfying the conditions in

Definition A.1.19, we identify the internal direct sum of Hi : 1 ≤ i ≤ n with their


external direct sumn⊕i=1

Hi via

h = h1 + h2 + · · ·+ hn ←→ (h1, h2, · · · , hn)

Definition A.1.20. For i = 1, 2, let Hi and Ki be Hilbert spaces, and φi ∈ End(Hi, Ki).

Then φ1 ⊕ φ2 : H1 ⊕H2 −→ K1 ⊕K2 is the endomorphism given by

φ1 ⊕ φ2 (h1, h2) = (φ1 (h1) , φ2 (h2))

for (h1, h2) ∈ H1 ⊕H2.

Let H and K be two vector spaces, consider the free vector space L = C(H×K).

By identifying H ×K as a subset of L, one considers in L the subspace N generated

by the elements of the form

(λ1h1 + λ2h2, µ1k1 + µ2k2)− λ1µ1(h1, k1)− λ1µ2(h1, k2)− λ2µ1(h2, k1)− λ2µ2(h2, k2)

The tensor product of H and K, denoted by H ⊗ K is defined to be the quotient

vector space L/N . For h ∈ H and k ∈ K, the element h⊗ k denotes the element of

H ⊗ K represented by the element (h, k) of L/N . The tensor product of H and K

can be abstractly characterized as following:

Definition A.1.21. Let H and K be two vector spaces. The tensor product of H

and K is a vector space H ⊗K with a bilinear map Θ : H ×K −→ H ⊗K which is

universal in the following sense, if Ψ : H×K −→ V is bilinear map into some vector

space V , then there exists a unique linear map T : H⊗K −→ V such that T Θ = Ψ.

We denote Θ(h, k) by h ⊗ k, so the map T is characterized by T (h ⊗ k) = Ψ(h, k).

The space H ⊗ C is identified with H via the unitary map h⊗ λ 7−→ λh.

Proposition A.1.22. Let H and K be two vector spaces. If e1, e2, ....en and

f1, f2, ...fm are bases for H and K respectively, then ei⊗fj : 1 ≤ i ≤ n, 1 ≤ j ≤ m

forms a basis for H ⊗K. In particular dim(H ⊗K) = dim(H) · dim(K).


Example A.1.23.

1. Cm ⊗ Cn ' Cmn.

2. For vector spaces H and K, we have End(H,K) ' K ⊗H∗.

According to the example above, the space End(H,K) is generated by xy∗ :

x ∈ K, y ∈ H, where y∗ denotes the linear form on H given by y∗(z) = 〈y |z 〉H

, and

xy∗ denotes the map xy∗(z) = 〈y |z 〉Hx for any z ∈ H.

Proposition A.1.24. For i = 1, 2, let Hi and Ki be vector spaces. If S ∈ End(H1, H2)

and T ∈ End(K1, K2), then there exists a unique linear map denoted S ⊗ T from

H1 ⊗K1 to H2 ⊗K2 characterized by S ⊗ T (h ⊗ k) = S(h) ⊗ T (k). If S and T are

isomorphisms, then S ⊗ T is an isomorphism with inverse S−1 ⊗ T−1.

Proposition A.1.25. For i = 1, 2, let Hi and Ki be vector spaces. There exist a

linear isomorphism T : End(H1, H2) ⊗ End(K1, K2) −→ End(H1 ⊗ K1, H2 ⊗ K2)

given by

T (A⊗B)(h1, k1) = A(h1)⊗B(k1)

for A⊗B ∈ End(H1, H2)⊗ End(K1, K2), and extending linearly.

Lemma A.1.26. Let H and K be inner product spaces whose inner products are

given by 〈· |·〉H

and 〈· |·〉K

respectively. There exists a unique inner product on H⊗K

such that 〈h1 ⊗ k1 |h2 ⊗ k2 〉H⊗K= 〈h1 |h2 〉H 〈k1 |k2 〉H. If H and K are Hilbert spaces,

then H ⊗K is a Hilbert space.

For the following proposition see [3, p.69, p.74].

Proposition A.1.27. Let (H, πH) and (K, πK) be two representations of a group G.

The tensor product πH ⊗ πK (resp. the direct sum πH ⊕ πK) defines a representation

of G in H ⊗K (resp. H ⊕K) given by

g 7−→ πH(g)⊗ πK(g)


(resp. g 7−→ πH(g)⊕ πK(g))

A.2 Positive and completely positive maps

Definition A.2.1. Let H and K be two Hilbert spaces. A linear map Φ : End(H) −→

End(K) is

• positive if Φ(A) ≥ 0 for any positive A ∈ End(H).

• n-positive if Φ⊗ In is a positive map, where Φ⊗ In denotes the linear map from

End(H)⊗Mn into End(K)⊗Mn, such that

Φ⊗ In (A⊗B) = Φ(A)⊗B

for all A ∈ End(H) and B ∈Mn.

• completely positive if it is n-positive for each n ≥ 1.

Before stating the next proposition, we introduce the following maps that are

needed for the proof.

Notation A.2.2. For k ≥ 2,

• let ιk denote the canonical inclusion of Ck−1 in Ck ' Ck−1 × C given by

ιk : Ck−1 −→ Ck

x 7−→ (x, 0)

for x ∈ Ck−1. The adjoint map of ιk is the projection ι∗k : Ck −→ Ck−1 that

maps the first k − 1 component of x ∈ Ck to themselves, and the last one to

the zero.


• For a Hilbert space H, let

– σH denote the map IH ⊗ ιk : H ⊗ Ck−1 −→ H ⊗ Ck.

– σ∗H denote the adjoint map of σH, IH ⊗ ι∗k : H ⊗ Ck −→ H ⊗ Ck−1.

Proposition A.2.3. Let H and K be Hilbert spaces, and Φ : End(H) −→ End(K)

be a linear map. For k ≥ 2, if Φ is k-positive, then it is (k − 1)-positive.

Proof:

Assume that Φ is k-positive. By Proposition 3.2.13, using the same notation in

A.2.3, we have

AdσH : End(H)⊗Mk−1 −→ End(H)⊗Mk

Adσ∗K : End(K)⊗Mk −→ End(K)⊗Mk−1

and

Φ⊗ Ik : End(H)⊗Mk −→ End(K)⊗Mk

are positive maps, so is the maps composition Adσ∗K (Φ⊗ Ik) AdσH : End(H) ⊗

Mk−1 −→ End(K)⊗Mk−1 which Φ⊗ Ik−1.

Proposition A.2.4. Let H and K be Hilbert spaces, and Φ : End(H) −→ End(K) be

a linear map. The map Φ is positive if and only if 〈Φ(A) |B 〉End(K)

≥ 0 for A ∈ End(H)

and B ∈ End(K) such that A,B ≥ 0.

Proof:

Let A ∈ End(H) and B ∈ End(K) such that A,B ≥ 0. Assume that Φ is

positive, then Φ(A) ≥ 0. Thus, there exist X, Y ∈ End(K) such that Φ(A) = XX∗

and B = Y Y ∗. Hence,

〈Φ(A) |B 〉End(K)

= 〈XX∗ |Y Y ∗ 〉End(K)

= tr(XX∗Y Y ∗) = tr((X∗Y )∗X∗Y ) ≥ 0.


The other direction, follows by Proposition A.1.17.

Recall (by Definition 4.5.1) that the dual of a linear map Φ : End(H) −→

End(K) is the unique map Φ∗ : End(K) −→ End(H) such that

〈B |Φ(A)〉End(K)

= 〈Φ∗(B) |A〉End(H)

for all A ∈ End(H) and B ∈ End(K).

Proposition A.2.5. Let H and K be Hilbert spaces, and Φ : End(H) −→ End(K)

be a linear map. Then

1. Φ is a positive map if and only if Φ∗ is positive.

2. Φ is completely positive if and only if Φ∗ is completely positive.

3. Φ∗ is trace preserving map if and only if Φ(IH) = IK.

Proof:

The proof of (1) and (2) follows directly from Proposition A.2.4, as (Φ⊗ In)∗ =

Φ∗ ⊗ In for each n. For (3), note that for any T ∈ End(K), we have

tr(Φ∗(T )) = 〈IH |Φ∗(T )〉End(H)

= 〈Φ(IH) |T 〉End(K)

and

tr(T ) = 〈IK |T 〉End(K)

Hence, Φ∗ is trace preserving if and only if 〈Φ(IH) |T 〉End(K)

= 〈IK |T 〉End(K)for each

T ∈ End(K). i.e. if and only if Φ(IH) = IK.

Proposition A.2.6. Let (H, πH) and (K, πK) be two unitary representations of a

group G. A linear map Φ : End(H) −→ End(K) is G-equivariant if and only if Φ∗

is G-equivariant.


Proof:

Assume Φ is G-equivariant. By Proposition 1.2.4,

Φ(πH(g)Aπ∗H(g)) = πK(g)Φ(A)π∗K(g) ∀g ∈ G

i.e.

Φ AdπH (g) = AdπK (g) Φ ∀g ∈ G

hence

Φ∗ AdπK (g) = AdπH (g) Φ∗ ∀g ∈ G

i.e

Φ∗ is G-equivariant.

As (Φ∗)∗ = Φ, the other direction also holds.

A.3 Covariant Stinespring dilation theorem

In this section, we state and give a proof of a special case of the covariant Stinespring

dilation theorem. The general case can be proved with some modification. The

following is a special case of the Stinespring dilation theorem [28, p.43-45].

Theorem A.3.1. Let H be finite dimensional Hilbert spaces, and Φ : Mn (C) −→

End(H) be a completely positive map. Then there exists a Hilbert space K, a unital

∗-homomorphism π : Mn(C) −→ End(K), and a bounded operator V : H −→ K

with ‖Φ(In)‖ = ‖V ‖2 such that

Φ(A) = V ∗π(A)V ∀A ∈Mn (C)

Remark A.3.2. If G is a group and ρ : G −→ End(Cn) a representation of G in

Cn then Adρ : G −→ End(Mn) where Adρ(g) = Adρ(g) is a representation of G in

Mn (C) .


Following the proof’s steps of Stinespring dilation theorem in [28, p.43-45], we

give a proof of the following theorem :

Theorem A.3.3. (Covariant Stinespring dilation theorem ) Let n ∈ N, H be a finite

dimensional Hilbert space, (H, ρH) and (Cn, ρn) be two unitary representations of a

group G, and Φ : Mn (C) −→ End(H) be a completely positive G-equivariant map.

Then there exist

1. a finite dimensional Hilbert space K with a unitary representation σ of G in K,

2. a G-equivariant unital ∗−homomorphism π : Mn (C) −→ End(K), and

3. a G-equivariant bounded operator V : K −→ K such that

Φ(A) = V ∗π(A)V ∀A ∈Mn (C)

Proof:

We follow the scheme of the proof given in [28, p.43-45]. On the algebraic tensor

productMn(C)⊗H, we consider the pre-inner product 〈· |·〉 defined for A,B ∈Mn(C)

and x, y ∈ H by 〈A⊗ x |A⊗ y 〉 = 〈Φ(B∗A)x |y 〉H

where 〈· |·〉H

is the inner product

on H. As ΦAdρn(g) = AdρH(g) Φ, ∀g ∈ G, then for all A,B ∈Mn(C) and x, y ∈ H,

we have

⟨Adρn(g) ⊗ ρH(g) (A⊗ x)

∣∣Adρn(g) ⊗ ρH(g) (B ⊗ y)⟩

=⟨Adρn(g)(A)⊗ ρH(g) (x)

∣∣Adρn(g)(B)⊗ ρH(g) (y)⟩

=⟨Φ((Adρn(g)(B)

)∗Adρn(g)(A)

)ρH(g) (x) |ρH(g) (y)

⟩H

=⟨Φ(Adρn(g)(B

∗A))ρH(g) (x) |ρH(g) (y)

⟩H

= 〈ρH(g)Φ (B∗A) (ρH(g))∗ ρH(g) (x) |ρH(g) (y)〉H

= 〈ρH(g)Φ (B∗A) (x) |ρH(g) (y)〉H


= 〈Φ (B∗A) (x) |y 〉H

= 〈A⊗ x |B ⊗ y 〉 .

Therefore, the map 〈· |·〉 is G -invariant under the action of ρ = Adρn(g) ⊗ ρH. Thus,

the subspace (as checked in [28])

N = u ∈Mn(C)⊗H : 〈u |v 〉 = 0 ∀ v ∈Mn(C)⊗H

is G-invariant under ρ. Hence, the induced bilinear form on K = (Mn(C)⊗H) /N is

a G-invariant inner product with respect to the action of G on K given by

σ(g)(u+N ) = ρ(g)(u) +N .

For A ∈Mn(C), the linear map π(A) : Mn ⊗H −→Mn ⊗H defined by

π(A)(∑

Ai ⊗ xi)

=∑

AAi ⊗ xi

satisfies

(Adρ(g)π(A))(∑

Ai ⊗ xi)

= (ρ(g)π(A)ρ∗(g))(∑

Ai ⊗ xi)

= ρ(g)π(A)(∑

ρ∗n(g)Aiρn(g)⊗ ρ∗H (g) (xi))

= ρ(g)

(∑Aρ∗n(g)Aiρn(g)⊗ ρ∗H (g) (xi)

)=∑

ρn(g) (Aρ∗n(g)Aiρn(g)) ρ∗n(g)⊗ ρH (g)ρ∗H (g) (xi)

=(∑

ρn(g)Aρ∗n(g)Ai ⊗ xi)

= π Adρn(g)(A)(∑

ai ⊗ xi).

So, π is aG-equivariant map for the actionAdρn(g) onMn (C) andAdρ on End(Mn⊗H)

with % = Adρn(g) ⊗ ρH. As shown in [28, p.43-45] π(A) leaves N invariant and thus

extends to a bounded linear operator on K, which is G-equivariant. Then π : Mn −→

End(K) is a unital ∗-homomorphism such that for each A ∈ Mn (C), g ∈ G and

u+N ∈ K, we have

(Adσ(g)π(A)) (u+N ) = (σ(g)π(a)σ∗(g)) (u+N )


= ρ(g)π(A)ρ∗(g) (u) +N = π Adρn(g)(A)(u) +N

= π Adρn(g)(A)(u+N ).

So, Adσ(g) π = π Adρn(g), and π is G-equivariant for the actions Adρn(g) on Mn (C)

and Adσ on End(K). The map V : H −→ K defined by V (x) = In ⊗ x+N satisfies

V (ρH(g)(x)) = In ⊗ ρH(g)(x) +N

= Adρn(g)(In)⊗ ρH(g)(x) +N

= Adρn(g) ⊗ ρH(g)(In ⊗ x+N )

= σ(g)V (x)

As shown in [28, p.43-45], or by direct computation

Φ(A) = V ∗π(A)V

for each A ∈Mn (C) .

Appendix B

Deferred Proofs

B.1 Deferred proofs in chapter 2

The proof of Lemma 2.2.3

Recall that for m,n ∈ N, we identify the spaces Pm ⊗ Pn and Pm,n, and by Remark

2.2.1, we have

ρm(g)⊗ ρn(g)f(x, y) = f(ax1 − bx2, bx1 + ax2, ay1 − by2, by1 + ay2)

where f(x, y) := f(x1, x2, y1, y2) ∈ Pm ⊗ Pn, and g =[

a b

−b a

]∈ SU(2). Intuitively,

the action of SU(2) on any polynomial f(x, y) in Pm ⊗ Pn, is by replacing x1 by

ax1 − bx2, x2 by bx1 + ax2, y1 by ay1 − by2 and y2 by by1 + ay2. The next lemma is

direct computations on the standard bases elements for Pm−1 ⊗ Pn and Pm ⊗ Pn−1

Lemma B.1.1. For m,n ∈ N. Let Mxi : Pm−1 ⊗ Pn −→ Pm ⊗ Pn, and Myi :

Pm ⊗ Pn−1 −→ Pm ⊗ Pn denote the multiplication by xi and yi respectively. Then(∂∂xi

)∗= Mxi and

(∂∂yi

)∗= Myi for each i = 1, 2.

Recall the maps ∆xy , ∆yx , Γxy and Ωxy in Definition 2.2.2.

157

B. Deferred Proofs 158

Lemma B.1.2. The operators ∆xy , ∆yx , Γxy and Ωxy are SU(2)-equivariant, and

satisfy

∆∗xy = ∆yx, Γ∗xy = Ωxy

Proof:

Let x := (x1, x2), y = (y1, y2), f(x, y) ∈ Pm ⊗ Pn, and g =[

a b

−b a

]∈SU(2).

Using Remark 2.2.1, we get

∂

∂x1

(ρm(g)⊗ ρn(g)f(x, y)) = a∂

∂x1

(ρm(g)⊗ ρn(g)f(x, y)) + b∂

∂x2

(ρm(g)⊗ ρn(g)f(x, y))

and

∂

∂x2

(ρm(g)⊗ ρn(g)f(x, y)) = −b ∂

∂x1

(ρm(g)⊗ ρn(g)f(x, y)) + a∂

∂x2


Hence,

∆yx(ρm(g)⊗ ρn(g)f(x, y)) = y1∂

∂x1

(ρm(g)⊗ ρn(g)f(x, y)) + y2∂

∂x2


= (ay1 − by2)∂

∂x1

(ρm(g)⊗ ρn(g)f(x, y)) + (by1 + ay2)∂

∂x2


= (ρm−1(g)⊗ ρn+1(g))

(y1

∂

∂x1

f ((x, y)) + y2∂

∂x2

f ((x, y))

)= (ρm−1(g)⊗ ρn+1(g)) ∆yx(f(x, y)).

i.e. ∆yx is SU(2)-equivariant.

By Lemma B.1.1, we have

〈∆xy (f(x, y)) |g(x, y)〉Pm+1⊗Pn−1

= 〈f(x, y) |∆yx (g(x, y))〉Pm⊗Pn

for g(x, y) ∈ Pm+1 ⊗ Pn−1, then ∆∗yx = ∆xy . By Proposition 1.2.7, the map ∆xy is

also SU(2)-equivariant.

Similarly, we have:

(ρm+1(g)⊗ ρn+1(g)) Γxy (f(x, y)) = (ρm+1(g)⊗ ρn+1(g)) ((x1y2 − y1x2) f(x, y))


=[(ax1 − bx2)(by1 + ay2)− (ay1 − by2)(bx1 + ax2)

]f(ax1− bx2, bx1 + ax2, ay1−

by2, by1 + ay2)

= (aax1y2 − bbx2y1 + bbx1y2 − aax2y1) (ρm(g)⊗ ρn(g)f(x1, x2, y1, y2))

= (aa+ bb)(x1y2 − x2y1) (ρm(g)⊗ ρn(g)f(x1, x2, y1, y2))

= det(g)(x1y2 − x2y1) (ρm(g)⊗ ρn(g)f(x, y))

= Γxy (ρm(g)⊗ ρn(g)f(x, y))

Thus Γxy is SU(2)-equivariant. By Lemma B.1.1, we have

〈Γxy (f(x, y)) |g(x, y)〉Pm+1⊗Pn+1

= 〈f(x, y) |Ωxy (g(x, y))〉Pm⊗Pn

for g(x, y) ∈ Pm+1 ⊗ Pn+1, which gives Γ∗xy = Ωxy. By Proposition 1.2.7, Ωxy is

SU(2)-equivariant.


Recall the isometry αm,n,h in Definition 2.3.2.

Lemma B.1.3. For m,n, h ∈ N with 0 ≤ h ≤ minm,n, let r = m+n− 2h. Using

the standard basis for Pk where k ∈ m,n, r, we have:

1. αm,n,h (f ri ) =h∑s=0



2. α∗m,n,h(fml ⊗ fnj

)=

(minh,j,m−l∑

s=max0,h−l,h+j−nβm,n,hl+j−h,s,j

)f rl+j−h if 0 ≤ l + j − h ≤ r

0 otherwise


where βm,n,hi,s,j = (−1)s(hs

)(n−hj−s

)(m−hi−j+s

)(m−h)!

√cm,n,hr! m! n!


Proof:

Recall that the standard basic element of Pk is f ki = aikxi1x

k−i2 where aik =

1√i!(k−i)!

.

As

∆n−hyx =

(y1

∂∂x1

+ y2∂∂x2

)n−h=

n−h∑t=0

(n−ht )yt1yn−h−t2

∂t

∂xt1

∂n−h−t

∂xn−h−t2

and

Γhxy = (x1y2 − y1x2)h =h∑s=0

(−1)s(hs) (x1y2)h−s (x2y1)s

by the definition of αm,n,h, we have:

For 0 ≤ i ≤ r

αm,n,h (f ri ) =√cm,n,ha

ir

h∑s=0

n−h∑t=0

(−1)s(hs)(n−ht )xh−s1 xs2y

(t+s)1 y

n−(s+t)2

∂t

∂xt1xi1

∂n−h−t

∂xn−h−t2

xr−i2

But

∂t

∂xt1xi1

∂n−h−t

∂xn−h−t2

xr−i2 =

i!(r−i)!

(i−t)!(m−h−i+t)! xi−t1 xm+t−h−i

2 −m+ i+ h ≤ t ≤ i

0 otherwise

Thus, we can rewrite the above sum as


ir

h∑s=0

mini, n−h∑t=max0,−m+i+h

U(m,n, h, i, s, t)xh−(s+t)+i1 x

m+(s+t)−h−i2 y

(t+s)1 y

n−(s+t)2

where U(m,n, h, i, s, t) = (−1)s(hs

)(n−ht

) i!(r−i)!(i−t)!(m−h−i+t)! .

Changing the summation variable in the inner sum to j = s+ t , we obtain


ir

h∑s=0


(−1)s(hs

) (n−hj−s

) i!(r−i)!(i−j+s)!(m−h−i+j−s)!x

lij1 x

m−lij2 yj1y

n−j2


Finally, asair

alijm a

jn

=√

j! lij !(m−lij)!(n−j)!i!(r−i)! , then

αm,n,h (f ri ) =√cm,n,h

h∑s=0


(−1)s(hs

)(n−hj−s

)(m−hi−j+s

)(m−h)!

√r! m! n!

(ri) (mlij) (nj)fmlij f

nj

=h∑s=0



In similar way, As

∆n−hxy =

n−h∑t=0

(n−ht

)xn−h−t1 xt2

∂n−h−t

∂yn−h−t1

∂t

∂yt2

and

Ωhxy =

h∑s=0

(−1)s(hs)∂h−s

∂xh−s1

∂s

∂xs2

∂h−s

∂yh−s2

∂s

∂ys1

then by Lemma 2.3.4, we have


)= √

cm,n,h almajn

n−h∑t=0

h∑s=0

(−1)s(hs) (n−ht )xn−h−t1 xt2∂h−s

∂xh−s1

(xl1) ∂s

∂xs2(xm−l2 ) ∂n−h−t+s

∂yn−h−t+s1

yj1∂h−s+t

∂yh−s+t2

yn−j2

= √cm,n,h alma

jn

h∑s=0

n−h∑t=0

(−1)s(hs) (n−ht )xn−h−t1 xt2∂h−s

∂xh−s1

(xl1) ∂s


∂yn−h−t+s1

yj1∂h−s+t

∂yh−s+t2

yn−j2

but

∂n−h−t+s

∂yn−h−t+s1

yj1∂h−s+t

∂yh−s+t2

yn−j2 =

j! (n−j)! s = t− n+ h+ j

0 otherwise

and

∂h−s

∂xh−s1

(xl1) ∂

s

∂xs2

(xm−l2

)=

l!

(l−h+s)!xl−h+s

1(m−l)!

(m−l−s)! xm−l−s2 h− l ≤ s ≤ m− l

0 otherwise

As for 0 ≤ t ≤ n− h we have h+ j − n ≤ s ≤ j, then


∂h−s

∂xh−s1

(xl1) ∂s


∂yn−h−t+s1

yj1∂h−s+t

∂yh−s+t2

yn−j2

=

j! (n−j)!l!(m−l)!

(l−h+s)!(m−l−s)! xl−h+s1 xm−l−s2 maxh− l, h+ j − n ≤ s ≤ minj,m− l

0 otherwise

As for h + j − n ≤ s ≤ j, the condition h − l ≤ s ≤ m − l is equivalent to

0 ≤ l + j − h ≤ r , we have

• If l and j satisfy 0 ≤ l + j − h ≤ r , then

Hence


)=√cm,n,h a

lma

jn

minh,j,m−l∑s=max0,h−l,h+j−n

(−1)s(hs) (n−hj−s)j! (n−j)!l!(m−l)!

(l−h+s)!(m−l−s)! xl+j−h1 xr−l−j+h2

=

(minh,j,m−l∑

s=max0,h−l,h+j−n

(−1)s

(n−hj−s

) (hs

) (m−hl−h+s

)(m−h)!

√cm,n,hr!m!n!(ml

)(nj

)(r

l+j−h

)) al+j−hr xl+j−h1 xr−l−j+h2

=

(minh,j,m−l∑

s=max0,h−l,h+j−n

βm,n,hl+j−h,s,j

)f rl+j−h.

• Otherwise, α∗m,n,h(fml ⊗ fnj

)= 0.

The proof of Corollary 2.3.15

Recall that

εji (m,n,h) =


βm,n,hi,s,j

where βm,n,hi,s,j = (−1)s(hs

)(n−hj−s

)(m−hi−j+s

)(m−h)!

√cm,n,hr! m! n!


.


Corollary B.1.4. For m,n, h ∈ N with 0 ≤ h ≤ minm,n, let r = m+n−2h. The

matrix coefficients εji := εji (m,n,h) of the isometry αm,n,h satisfy

1. εji = (−1)hεn−jr−i for any 0 ≤ i ≤ r and j ∈ B(i).

2. εi+hi = βm,n,hi,h,h+i for any i ≤ n− h.

3. For n− h ≤ i ≤ r, εni = βm,n,hi,h,n = (−1)h∣∣∣βm,n,hi,h,n

∣∣∣ 6= 0.

4. For j ∈ B(0), we have εj0 = βm,n,h0,j,j = (−1)j

∣∣∣βm,n,h0,j,j

∣∣∣ 6= 0.

5. For 0 ≤ i ≤ m− h we have ε0i = βm,n,hi,0,0 6= 0.

6. For j ∈ B(i), we have εji (m,n,n) = βm,n,ni,j,j and εji (m,n,0) = βm,n,0i,0,j

Proof:

(1) Let g0 =[

0 1

−1 0

]. By the SU(2)-equivariance of αm,n,h, we have

(ρm(g0)⊗ ρn(g0))αm,n,hρ∗r(g0)(f ri ) = αm,n,h(f

r

i )

for 0 ≤ i ≤ r.

As

• ρ∗r (g0) (f ri ) = (−1)r−if rr−i,

• ρm(g0) (fml ) = (−1)lfmm−l , and

• ρn(g0)(fnj)

= (−1)jfnn−j

using Corollary 2.3.13, the above equation can be written as

(−1)h

minr−i+h, n∑s=max0,−m+r−i+h

εsr−i fm

li(n−s)⊗ fnn−s =


εji fm

lij⊗ fnj

Let t = n− s.

Since n−max0,−m+ r − i+ h = mini+ h, n, and


n−minr − i+ h, n = max0,−m+ i+ h, we have:

(−1)h

mini+h, n∑t=max0,−m+i+h

εn−tr−i fm

lit⊗ fnt =


εji fm

lij⊗ fnj

By the linearly independence of the vectors fml ⊗ fnj , the last equality implies that

(−1)hεn−jr−i = εji for any j such that max0,−m+ i+ h ≤ j ≤ mini+ h, n.

The other statements follow by direct computations.


Lemma B.1.5. For m ∈ Nr 0.

m−1∑k=0

(m−1k

)2(mk+1

)(mk

) = m(m+1)(m+2)6m2

Proof:

m−1∑k=0

(m−1k

)2(mk+1

)(mk

) = 1m2

m−1∑k=0

(k+1)(m−k) = 1m2

m∑k=1

k(m− k + 1) = 1m2

[m∑k=1

k(m+ 1)−m∑k=1

k2

]Since

m∑k=1

k = m(m+1)2

, andm∑k=1

k2 = m(m+1)(2m+1)6

the result follows.

Recall that B(i) := j : max0,−m+ i+ h ≤ j ≤ mini+ h, n.

Lemma B.1.6. Let m ∈ N \ 0 and εji := εji (m,m,m−1). The following identities hold

1. εj0 = (−1)j

√6(j+1)(m−j)m(m+1)(m+2)

for 0 ≤ j ≤ m− 1.

2. εj1 = (−1)j√

3m(m+1)(m+2)

(m− 2j) for 0 ≤ j ≤ m.


3. εj2 = (−1)j−1

√6j(m−j+1)

m(m+1)(m+2)for 1 ≤ j ≤ m.

Proof:

1. For j ∈ B(0) = j : 0 ≤ j ≤ m− 1, using Lemma B.1.5, we have

εj0 =

minm−1,j,j+1∑s=max0,j,j−1

βm,m,m−10,s,j =

j∑s=j

βm,m,m−10,s,j = βm,m,m−1

0,j,j = (−1)j

√6(j+1)(m−j)m(m+1)(m+2)

2. For j ∈ B(1) = j : 0 ≤ j ≤ m, we have

εj1 =

minm−1,j∑s=max0,,j−1

βm,m,m−11,s,j =

βm,m,m−11,0,0 j = 0

βm,m,m−11,j−1,j + βm,m,m−1

1,j,j 1 ≤ j ≤ m− 1

βm,m,m−11,m−1,m j = m

by Lemma B.1.5, we get:

• βm,m,m−11,0,0 =

√3m

(m+1)(m+2).

• βm,m,m−11,j−1,j + βm,m,m−1

1,j,j = (−1)j√

3m(m+1)(m+2)

(m− 2j) for any 1 ≤ j ≤ m− 1.

• βm,m,m−11,m−1,m = (−1)m−1

√3m

(m+1)(m+2).

In all cases, we have εj1 = (−1)j√

3m(m+1)(m+2)

(m− 2j).

3. Similarly for j ∈ B(2) = j : 1 ≤ j ≤ m, we have

εj2 =

minm−1,j,j−1∑s=max0,,j−2,j−1

βm,m,m−12,s,j =

,j−1∑s=,j−1

βm,m,m−12,s,j = βm,m,m−1

2,j−1,j = (−1)j−1

√6j(m−j+1)

m(m+1)(m+2).


B.2 Proofs in chapter 5

The proof of Theorem 5.2.2

Recall the definition of direct sum of operator in Definition 5.2.1.

Theorem B.2.1. Let G be a group. Let H =r⊕t=1

Wt where Wt : 1 ≤ t ≤ r are

nonequivalent G-irreducible spaces, and K =m⊕s=1

Vs where Vs : 1 ≤ s ≤ m are

nonequivalent G-irreducible space. If there exist k ∈ N, such that

1. For each 1 ≤ t ≤ k , Wt ' Vt via a G-equivariant isomorphism

ψt : Wt −→ Vt

2. For each t > k the subspace Wt is not equivalent to any of the Vs for any

1 ≤ s ≤ m.

Then, for any G-equivariant map Φ : H −→ K, there exist λ1, λ2, ........., λk such

that Φ is the orthogonal direct sum of the operators λtψt : 1 ≤ t ≤ k

i.e.

Φ =k⊕t=1

λtψt

Proof:

By the assumption in (1) and (2), and since the multiplicities of the subspaces Wt

and Vs, in H and K are one, we have Wt Vs for any s 6= t. Let ιt, ιs and qt, qs denote

the inclusion maps, and the orthogonal projections of Wt and Vs respectively. As the

maps ιt, ιs,qt, qs are G-equivariant maps (Lemma 1.2.10), the maps composition

qsΦιt : Wt −→ Vs

is G-equivariant for any any 1 ≤ s ≤ m and 1 ≤ t ≤ r. By Schur’s Lemma 1.2.12,

the map qsΦιt is a zero map for s 6= t.


If s = t, then by Schur’s Lemma a gain, the map ψ−1t qtΦιt : Wt −→ Wt is multiple of

the identity. i.e. ψ−1t qtΦιt = λtIWt for some λt ∈ C. That is

Φ |Wt = λtψt

For u ∈ H =r⊕t=1

Wt, we have

Φ(u) = Φ(u1, u2, ....ur) ut ∈ Wt

Thus

Φ(u) = (λ1ψ1(u1), λ2ψ2(u2), ...., λkψk(uk), 0, .., 0)

B.3 Proofs in Chapter 7


Recall that for m ∈ N r 0, the EPOSIC Kraus channel Φm,1,1 : End(Pm−1) −→

End(Pm) has two Kraus operator T0, T1 (Definition 4.2.1). For a pure state % = ww∗

in End(Pm−1), the set UΦm,n,n,% = u0, u1 where uj = Tjw , j = 1, 2, by Remark 6.1.8,

we have Φm,1,1(ww∗) =

i∑j=0

uju∗j . The following lemma follows by direct computations

using the formula εji (m,n,h) =minh,j,j+m−i−h∑s=max0,j−i,j+h−n

βm,n,hi,s,j given in Corollary 2.3.13. Item (3)

follows as Φm,1,1 is trace preserving.

Lemma B.3.1. For m ∈ Nr 0 and εji := εji (m,1,1), we have

1. ε0l =

√l+1m+1

, ε1l = −

√m−lm+1

, and (ε0l )

2 + (ε1l )

2 = 1, for 0 ≤ l ≤ m− 1.

2. (ε0l )

2 = (ε0l−1)

2 + 1m+1

, and (ε1l−1)

2 = (ε1l )

2 + 1m+1

, for 1 ≤ l ≤ m− 1.


3. ‖u0‖2 + ‖u1‖2 = 1.

Following the same notations in Proposition 7.2.3, we have

Lemma B.3.2. Let m ∈ Nr0. The minimal value of R = ‖u0‖2 ‖u1‖2−|〈u0 |u1 〉|2

is m(m+1)2 .

Proof:

Let w =m−1∑l=0

wlfm−1l ∈ Pm−1 be a unit vector. By Lemma 7.1.5, we have

u0 =m∑l=1

ε0l−1wl−1f

ml and u1 =

m−1∑l=0

ε1lwlf

ml .

Thus

〈u0 |u1 〉 =m−1∑l=1

ε0

l−1wl−1ε1

lwl =m−1∑l=1

ε0

l−1wlε1

lwl−1 = 〈v0 |v1 〉

where v0 =m−1∑l=1

ε0l−1wlf

ml , and v1 =

m−1∑l=1

ε1lwl−1f

ml .

Using Lemma B.3.1, we obtain

‖v0‖2 =m−1∑l=1

(ε0

l−1

)2 |wl|2 =m−1∑l=1

(ε0

l−1

)2 |wl|2 +‖w‖2

m+1− ‖w‖

2

m+1

= 1m+1|w0|2 +

m−1∑l=1

((ε0l−1

)2+ 1

m+1

)|wl|2 −

‖w‖2

m+1

= (ε00)

2 |w0|2 +m−1∑l=1

(ε0l )

2 |wl|2 − 1m+1

Note that ‖w‖2 =m−1∑l=0

|wl|2 = 1.

Thus

‖v0‖2 =m−1∑l=0

(ε0

l )2 |wl|2 − 1

m+1=

m∑l=1

(ε0l−1

)2 |wl−1|2 − 1m+1

= ‖u0‖2 − 1m+1

Similarly ‖v1‖2 = ‖u1‖2 − 1m+1

. So

|〈u0 |u1 〉|2 = |〈v0 |v1 〉|2 ≤ ‖v0‖2 ‖v1‖2 =(‖u0‖2 − 1

m+1

) (‖u1‖2 − 1

m+1

)


= ‖u0‖2 ‖u1‖2 − 1m+1

(‖u0‖2 + ‖u1‖2)+ 1

(m+1)2 = ‖u0‖2 ‖u1‖2 − m(m+1)2

Thus

R = ‖u0‖2 ‖u1‖2 − |〈u0 |u1 〉|2 ≥ m(m+1)2

and m(m+1)2 is a lower bound for R. To verify that m

(m+1)2 is the minimal value of R,

compute R at w0 = (1, 0, . . . , 0)t to get Rw0 = m(m+1)2 .

Appendix C

List of Equations That Are Used in

The Computation

For m,n, h ∈ N with 0 ≤ h ≤ minm,n, let r = m+ n− 2h then

• cm,n,h =((m−h)!)2

r! m! n!

(h∑j=0

(hj

)2

( mh−j) (nj)

)

• βm,n,hi,s,j = (−1)s(hs

)(n−hj−s

)(m−hi−j+s

)(m−h)!

√cm,n,hr! m! n!


.

• εji (m,n,h) =minh,j,j+m−i−h∑s=max0,j−i,j+h−n

βm,n,hi,s,j .

• lij = h+ i− j.

• For 0 ≤ i ≤ r, B(i) = j : max0,−m+ i+ h ≤ j ≤ mini+ h, n.

170

C. List of Equations That Are Used in The Computation 171

•fml = almx

l1x

m−l2 : 0 ≤ l ≤ m

where alm =

1√l!(m−l)!

.

• Jm(fml ) = (−1)lfmm−l and J∗m(fml ) = (−1)m−lfmm−l.

• Pm ⊗ Pn 'minm,n⊕h=0

Pm+n−2h.

• αm,n,h (f ri ) =∑

j∈B(i)

εji (m,n,h) fmlij ⊗ fnj .

• ηm,n,h = (IPm ⊗ Jn)αm,n,h : Pm+n−2h −→ Pm ⊗ P n.

• α∗m,n,h(fml ⊗ fnj

)=

εjl+j−hf

rl+j−h if 0 ≤ l + j − h ≤ r

0 otherwise

.

• Φm,n,h(fri1f r∗i2

) =∑

j∈B(i1)∩B(i2)

εji1εji2fmli1j

fm∗

li2j.

• C(Φm,n,h) =r∑

i1,i2=0

( ∑j∈B(i1)∩B(i2)

εji1εji2fmli1j

fm∗

li2j

)⊗ Ei1i2 .

• C(Φm,n,h) = r+1n+1

qm,r,m−h.

• λ2t =

r∑j=m−h

(−1)t+jεjt (r,r,r−t)(εr−j+hr−j (m,n,h))2

m!√cm,m,m−t

•m∑k=1

k = m(m+1)2

, andm∑k=1

k2 = m(m+1)(2m+1)6

.

C. List of Equations That Are Used in The Computation 172

•m∑k=1

k(m− k + 1) = m(m+1)(m+2)6

andm∑k=1

k(k + 1) = m(m+1)(m+2)3

.

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