Coherence of quantum channels

Chandan Datta,1, 2, ∗ Sk Sazim,3, † Arun K. Pati,3, ‡ and Pankaj Agrawal1, 2, §

1Institute of Physics, Sachivalaya Marg, Bhubaneswar 751005, Odisha, India.2Homi Bhabha National Institute, Training School Complex, Anushakti Nagar, Mumbai 400085, India.

3QIC group, Harish-Chandra Research Institute, HBNI, Allahabad 211019, India

We investigate the coherence of quantum channels using the Choi-Jamiołkowski isomorphism. The relationbetween the coherence and the purity of the channel respects a duality relation. It characterizes the allowedvalues of coherence when the channel has certain purity. This duality has been depicted via the Coherence-Purity (Co-Pu) diagrams. In particular, we study the quantum coherence of the unital and non-unital qubitchannels and find out the allowed region of coherence for a fixed purity. We also study coherence of differentincoherent channels, namely, incoherent operation (IO), strictly incoherent operation (SIO), physical incoherentoperation (PIO) etc. Interestingly, we find that the allowed region for different incoherent operations maintainthe relation PIO ⊂ SIO ⊂ IO. In fact, we find that if PIOs are coherence preserving operations (CPO), itscoherence is zero otherwise it has unit coherence and unit purity. Interestingly, different kinds of qubit channelscan be distinguished using the Co-Pu diagram. The unital channels generally do not create coherence whereassome nonunital can. All coherence breaking channels are shown to have zero coherence, whereas, this is notusually true for entanglement breaking channels. It turns out that the coherence preserving qubit channels haveunit coherence. Although the coherence of the Choi matrix of the incoherent channels might have finite values,its subsystem contains no coherence. This indicates that the incoherent channels can either be unital or nonunitalunder some conditions. We also prove a complementarity relation between the relative entropy of coherenceand the Holevo quantity of the quantum channel. This suggests that the coherence and the Holevo quantity ofthe channels cannot be arbitrarily large at the same time.

I. INTRODUCTION

Quantum coherence and entanglement are two fundamen-tal resources in quantum information and computation [1–4].These two resources are closely related [5, 6]. While the con-cept of entanglement requires at least two particles, coherencecan be defined for a single system. Recent developments showthat coherence in a quantum system can be a useful resource inquantum algorithm [7–11], quantum meteorology [12], quan-tum thermodynamics [13–19], and quantum biology [20–22].Therefore, the study of resource theory of quantum coherenceis of immense importance [23–53].

The quantum coherence like other quantum resources isalso fragile in the presence of noisy environment. The in-teraction of quantum systems with environment have been ex-tensively studied using different models – in particular usingnoisy channels [54]. Characterizing all these channels andtheir effect on various physical resources are vital [54, 55].These channels are also important to construct resource theo-retic aspect of coherence. Here, in this work, we ask a reversequestion. Can we associate coherence with a quantum chan-nel? We answer this question positively. We define the coher-ence of quantum channels using the Choi-Jamiołkowski (C-J)isomorphism [56, 57]. In this paper, we consider the unitalas well as non-unital qubit channels [55, 58, 59]. We com-pute their coherence and purity analytically. While coherenceof a non-unital channel can go up to

√2 as measured by the

l1-norm, the coherence of unital channels can never exceed 1.

∗ chandan@iopb.res.in† sk.sazimsq49@gmail.com‡ akpati@hri.res.in§ agrawal@iopb.res.in

Using the coherence-purity (CoPu) diagrams, we find that itmay be possible to distinguish unital channels and non-unitalchannels.

The resource theory of coherence require two important el-ements – free states and free operations [3, 23]. Free statesare those which have no coherence in a given reference basis.Free operations do not create any coherence and are known asincoherent operations. Depending on the restrictions (physicalrequirement), there exist different types of incoherent opera-tions. The largest set of incoherent operations contains Max-imally Incoherent Operations (MIO) [60]. The other candi-dates are Incoherent Operations (IO) [23], Strictly Incoher-ent Operations (SIO) [4, 24], Physical Incoherent Operations(PIO) [28] etc. There are many other free operations in theliterature like Fully Incoherent Operations (FIO), Genuine In-coherent Operations (GIO) etc. It is an important task to un-derstand these operations and distinguish them. In this work,we aim to distinguish these operations using CoPu diagrams.A PIO is in fact a strange candidate – some PIOs which are notCoherence Preserving Operations (CPO) [38] have coherencezero but otherwise it has unit purity and unit coherence. Wehave constructed all possible FIOs for qubit case and showthat they have zero coherence when only row elements arenonzero in their Kraus representation. The FIOs which arediagonal are called GIOs [61]. The FIOs with anti-diagonalelements in their Kraus representation have same CoPu dia-grams as GIOs.

We also consider the class of coherence non-generatingqubit channels (CNC) as well as the channels to create maxi-mal coherence (CMC). CNC is the bigger set in comparison toall incoherent operations [62]. We also consider other knownqubit channels like the class of Pauli channels, degradable andanti-degradable channels, amplitude damping channels, depo-larizing channels, and homogenization channels and show that

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they might be distinguished using CoPu diagrams. Followingare the salient features of our results which we address exten-sively in the main text.

• The coherence of quantum channels has been identifiedwith the coherence of the Choi matrix. The characteri-zation of coherence quantifies the quantumness of thechannels, i.e., it might be considered as the quantitywhich characterizes how much the map is quantum.

• The relation between the coherence and the purity of thechannel respects a duality type relation. It characterizesthe allowed values of coherence while the channel hascertain purity. This duality has been depicted via theCoherence-Purity (Co-Pu) diagrams.

• Different kinds of qubit channels can be distinguishedusing the Co-Pu diagrams. For example, the unital andnonunital channels, the incoherent channels, degradableand anti-degradable channels etc can be distinguishedwith the help of our formalism.

• Unital channels generally do not create coherencewhereas some nonunital can.

• a) All coherence breaking channels have zero coher-ence. However, this is not usually true for entanglementbreaking channels. b) Moreover, the coherence preserv-ing qubit channels have unit coherence.

• Although the coherence of the Choi matrix of the inco-herent channels might have finite values, its subsystemcontains no coherence. This very fact tells us that the in-coherent channels can either be unital or nonunital with~τ = 0, 0, τz. (~τ is defined in Section II-A.)

• There exists a complimentarity relation between the co-herence and the Holevo quantity of the channel. Thismeans that the more coherence the channel contains, theless will be its Holevo quantity.

The paper is organized as follows. In Section II, we intro-duce some extant concepts relevant to our work. Section IIIcontains the detailed analysis of coherence content of all qubitunital as well as non-unital channels. Section IV presentsthe CoPu diagrams of the incoherent operations introducedrecently in the literature. In Section V, we discuss CoPu di-agrams of other relevant qubit channels. In Section VI, wediscuss the relationship between the relative entropy of coher-ence and Holevo quantity. Finally, we conclude in the lastsection.

II. PRELIMINARIES

Here, we will discuss some relevant concepts which are im-portant in explaining our main results.

A. Quantum Channels

A quantum channel is a completely positive and trace pre-serving (CPTP) linear map which maps a density matrix toa density matrix [54]. If a map, Φ, is CPTP, then it can berepresented by a set of Kraus operators Ki; i = 1, 2, ..., n

Φ[ρ] :=

n∑i

KiρK†i , (1)

with∑ni K

†iKi = 1. Here ρ is an arbitrary density matrix.

We call this representation as the Kraus representation of thechannel (KROC). However, in this work, we will mainly focuson qubit channels [58, 59].

The action of a qubit channel Φ can also be completelycharacterized by a 3 × 3 real matrix M and a 3-dimensionalvector ~τ [55, 58, 59]. An arbitrary qubit is expressed asρ = 1

2 (I + ~r · ~σ), where ~r is the 3-dimensional Bloch vector.The action of qubit channel Φ on ρ is described in followingway:

(1, ~r′)T = ΛΦ(1, ~r)T , (2)

where ΛΦ represents a real 4× 4 matrix and T denotes trans-position. The most general form of ΛΦ for complete positivitycan be written as

ΛΦ =

(1 01×3

~τ M

). (3)

It leads to the affine transformation of the Bloch vector, i.e.,~r′ = M~r+~τ . Up to some local unitary equivalence, any qubitchannel can be written as

ΛΦ =

1 0 0 0τx λx 0 0τy 0 λy 0τz 0 0 λz

, (4)

where λ’s are the (signed) singular values of the matrixM andτ ’s represents the shift of the coordinates [58, 59]. A channelis unital if and only if ~τ = 0.

The above representation of a qubit channel Φ is known asthe affine representation of the channel (AROC).

B. Choi-Jamiołkowski Isomorphism

In this paper, we wish to characterize a quantum channelby its coherence. To do this, we will use the idea of Choi-Jamiołkowski isomorphism [56, 57]. It permits one to asso-ciate a CPTP map Φ to a density matrix of composite systemAB with B being the auxiliary system of same dimension asA. The prescription is:

ρAB = Φ⊗ IB(|Ψ〉AB〈Ψ|), (5)

where |Ψ〉AB is a maximally entangled state. It states that forevery quantum state there is a unique quantum operation. It

3

is also known as the channel-state duality. As there is one toone map between the state and the channel, the coherence ofthe final state ρAB can represent the coherence of the quan-tum channel. If ρAB is separable then the channel, Φ is calledentanglement breaking channel. If the state ρAB is incoher-ent, the corresponding channel is called coherence breakingchannel. The density matrix ρAB is called the Choi matrix.

For qubit channels, without loss of generality, we will con-sider the singlet state as the two qubit maximally entangledstate. The canonical form of singlet state is

|Ψ〉AB〈Ψ| =1

4(I⊗ I−

3∑i=1

σi ⊗ σi), (6)

where σi (i = 1, 2, 3) are the Pauli matrices. Coherence of astate depends on the reference basis used to write it. Here andbelow, we shall use computational basis as reference basis.

C. Quantum coherence

Quantum coherence is a fundamental concept in quantummechanics. It arises due to the superposition principle [60].Recently, much attention has been paid to define proper mea-sure of quantum coherence [23, 28, 35, 61]. As coherence is abasis dependent quantity, we should first fix a particular basis.Let |i〉 (i = 1 . . . d) is a basis in a d-dimensional HilbertspaceHd. The density matrices which are diagonal in this ba-sis are called incoherent states. The structure of these densitymatrices is as follows

δ =

d∑i=1

δi|i〉〈i|, (7)

where∑di=1 δi = 1. Quantum operations with Kraus oper-

ators, Ki, satisfying∑iK†iKi = I, will be incoherent if

it takes an incoherent state to another incoherent state, i.e.,KiIK†i ∈ I for all i, where I is the set of all incoherentstates.

Any proper measure of the coherence C must satisfy thefollowing conditions [3, 23]:(C1) C(δ) = 0, where δ ∈ I. Hence, for any quantum stateC(ρ) > 0.(C2) It should not increase under any incoherent operation,i.e., C(ρ) > C(Φ[ρ]), where Φ[ρ] is any incoherent operation.(C3) C(ρ) is nonincreasing under selective measurements onaverage, C(ρ) >

∑i qiC(ρi), where qi = Tr(KiρK

†i ) and

ρi = KiρK†i /qi for all iwith

∑iK†iKi = I andKiIK†i ∈ I.

(C4) C(ρ) does not increase under mixing of quantum states,∑i piC(ρi) > C(

∑i piρi), with ρ =

∑i piρi.

The l1-norm of coherence, Cl1(ρ), and the relative entropyof coherence, Cr(ρ), satisfy all these conditions. The l1-normof coherence measure is defined as

Cl1(ρ) =∑i 6=j

|ρi,j |, (8)

where ρi,j = 〈i|ρ|j〉 and the relative entropy of coherence isdefined as

Cr(ρ) = S(ρD)− S(ρ), (9)

where ρD is the matrix constructed from ρ by removing all theoff-diagonal elements and S(ρ) represents the von Neumannentropy for the density operator ρ. This measure of quantumcoherence satisfies all the criteria as required to be a goodmeasure of coherence [3].

III. COHERENCE OF THE CHANNELS

Here we investigate the coherence of the unital as well asnon-unital channels. According to C-J isomorphism, the co-herence of the channels is equivalent to the coherence of thetransformed singlet state. Hence, the channel coherence andits other properties can easily be evaluated. In this section, wewill mainly follow the AROC.

A. Coherence of Unital Channels

Unital qubit channels are those which do not change themaximally mixed state, I/2. They satisfy,

∑iK†iKi = I =∑

iKiK†i . From the section II.B, it is clear that the set of uni-

tal channels can be represented as a three-parametric familyof completely positive maps. Now if we apply the C-J map onthe state given in (6), then the final state will be

ρAB =1

4

(I⊗ I− ~λ · (~σ ⊗ ~σ)

). (10)

The positivity of the eigenvalues of ρAB will ensure the com-plete positivity of the unital map. Let us define qij = 1 +(−1)iλx + (−1)i+jλy + (−1)jλz with i, j = 0, 1, where qijare the four eigenvalues of the density matrix ρAB . Therefore,the positivity constraints on the unital channels are [55]

qij ≥ 0. (11)

Using the l1-norm, the coherence of the unital channel is givenby

Cl1 =1

2

(|λx + λy|+ |λx − λy|

). (12)

Note that the coherence does not depend on λz . It implies thatmany isocoherence planes will lie along λz axis. This is theconsequence of the choice of reference basis. The coherenceof unital channel will reach its maximum value 1 when wehave λx = λy = ±1 or λx = −λy = ±1. The purity, asdefined by P = Tr[ρ2], of the unital channel is given by

P =1

4(1 + |~λ|2). (13)

It is well known that the unital channels form a tetrahedronwith unitary operators in its vertices [69]. The state ρAB

4

has the same geometrical picture. Bell states sits on the fourextremal points of the tetrahedron [69]. From eq. (13) itis clear that |~λ|2 = 4P − 1. Values of λi lie on the sur-face of sphere with the radius 4P − 1 centered at the pointλx = λy = λz = 0. The channels with the same purity forma sphere. Therefore, there may exist quantum channels withdifferent coherence but the same purity (see Fig.1).

Co

her

ence

0.25

0.5

0.75

1

1.251.5

Purity0.2 0.4 0.6 0.8 1

FIG. 1: (Color online) The figure depicts the allowed regionof coherence as measured by the l1-norm for unital (regioninside red curve) and non-unital (region inside black curve)

channels, respectively for the allowed purity range. TheCoPu diagram shows that the channels outside the overlap

region are non-unital and can be exactly distinguished fromunital ones. The purity for these channels, P ∈ [ 1

4 , 1].

B. Coherence of Non-unital Channels

The non-unital qubit channels are characterized by six pa-rameters as shown in the section II-A. The Choi matrix corre-sponding to the non-unital channels is given by

ρAB =1

4

((I + ~τ · ~σ)⊗ I− ~λ · (~σ ⊗ ~σ)

). (14)

The positivity of the non-unital channel is guaranteed byρAB ≥ 0. Let us define τ =‖ ~τ ‖ and n = ~τ

τ . Then thenon-unital map is positive iff

qij ≥ 0 and τ2 ≤ u−√u2 − q, (15)

where u = 1−∑λ2i + 2

∑λ2in

2i and q =

∏qij [55].

The coherence of the non-unital channel is given by

Cl1 =1

2

(|λx + λy|+ |λx − λy|+ 2

√τ2x + τ2

y

). (16)

Note that the coherence is independent of both λz and τz .Hence, some isocoherence planes will lie on the λz and τz

planes. The purity for the channel is given by

P =1

4(1 + |~λ|2 + |~τ |2). (17)

Eq. (17) can be written as |~λ|2 + |~τ |2 = 4P − 1. It is clear, asbefore, that for a fixed purity, the values of parameters char-acterizing non-unital channels lie on the surface of a sphere.By fixing purity we can get the allowed regions of coherenceas is shown in the Fig.1.Observation 1: If the coherence of the channel is more than1, then it is non unital. One can easily see this from Fig.1. Hence, CoPu diagrams can help us distinguishing betweenunital and non unital channels for some region.Observation 2: Unital channel cannot create coherence in thesubsystem A whereas the non-unital channel can.

It can be easily checked by looking at the density matrixof the subsystem A after the operation of the channel on thestate (6). For unital and non-unital channel density matricesof subsystem A are respectively

ρuA =1

2I and ρnuA =

1

2

(1 + τz τx − iτyτx + iτy 1− τz

). (18)

where ρuA = Tr[ΦuA ⊗ IB(|Ψ〉AB〈Ψ|)] and ρnuA = Tr[ΦnuA ⊗IB(|Ψ〉AB〈Ψ|)]. Note that the non-unital channels with ~τ =(0, 0, τz), can not create coherence in subsystem A.

If we look closely at Eq.(16) and Eq.(18), it is clear thatthe coherence of the nonunital channel can exactly be decom-posed into the coherence of unital channel plus the coherenceinduced in the subsystem A by the nonunital channel, i.e.,

Cl1(Φnu) = Cl1(Φu) + Cl1(ρnuA ). (19)

Also the Eq.(19) tells that Cl1(Φnu) ≥ Cl1(Φu). The Eq.(19)has been depicted in Fig.2. Plot shows that the minimum co-herence of nonunital channels are exactly equal to the coher-ence induced in the subsystem A.Proposition 1: All coherence breaking channels have zerocoherence.Proof. A quantum channel is called coherence breaking chan-nel if it maps any state to an incoherent state [63]. For a co-herence breaking channel, the ΛΦ should take the followingform [63]

ΛΦ =

1 0 0 00 0 0 00 0 0 0τz 0 0 λz

.

Applying this channel on the state (6), one can show that Choimatrix is

1

4

1 + τz − λz 0 0 00 1 + τz + λz 0 00 0 1− τz + λz 00 0 0 1− τz − λz

.

Clearly, we get an incoherent Choi matrix.However, note that this may not be the case for all entan-

glement breaking channels. It may happen that the final Choimatrix is separable but has coherence in subsystems.

5

Cl1(ρAB)

0

0.25

0.5

1

1.251.5

Cl1(ρA)

0 0.2 0.4 0.6 0.8 1

FIG. 2: (Color online) The channel coherence (Cl1(ρAB)) vscoherence induced in the subsystem A (Cl1(ρA)) plot for

nonunital qubit channels. The red curve depicts the nonunitalchannels which has maximum coherence for a given

subsystem coherence whereas blue one represents thenonunital channels with minimum coherence. Plot shows that

the minimum channel coherence is exactly equal to thecoherence induced in the subsystem A.

IV. COHERENCE OF INCOHERENT CHANNELS

The concept of incoherent operations is not unique. Thereexist many classes of incoherent operations in the literature.Below we consider the most important classes of incoherentoperations that have been discussed in the different resourcetheoretic perspectives of the quantum coherence.

Maximally Incoherent operation (MIO) : A channel Φ isMIO iff Φ[δ] ∈ I, for all incoherent states δ, i.e., MIO pre-serves the set of incoherent states. This is the largest set ofoperations which preserve incoherence [60].

Incoherent operation (IO) : In Ref. [23], a smaller and rele-vant class of incoherent operations was introduced. A channel

Φ with Kraus decomposition Ki is IO iff KiδK†i

Tr[KiδK†i ]∈ I for

all i and δ ∈ I. Hence, the Kraus operators of IO may beexpressed as

Ki =

d−1∑j=0

cij |fi(j)〉〈j|, (20)

where fi : 0, 1, .., d − 1 7→ 0, 1, .., d − 1 and d is thedimension of the Hilbert space. Note that coherence cannot begenerated, even probabilistically, from incoherent states dueto the action of this channel.

The above two incoherent operations are defined in termsof their inability to create coherence. One can add furtherdesirable restriction to the set of free operations. One such

constraint is that the operations will be unable to use the co-herence of the input state.

Strictly Incoherent operation (SIO) : A channel Φ is SIO iffits Kraus operators Ki individually commutes with dephas-ing, i.e., 4(KiδK

†i ) = Ki4(δ)K†i , where 4 is dephasing

operation [4, 24] defined as 4(ρ) =∑i〈i|ρ|i〉|i〉〈i|. This

condition makes fi one-to-one, i.e., fi becomes permutation,πi in Eq.(20). Thus, SIO admits the set Ki as well asK†i arealso incoherent. This indicates that the SIOs are not capableof using coherence of initial input states [24].

The above mentioned operations cannot be implemented byintroducing an incoherent environment and a global unitaryoperation. This observation led one to introduce physicallymotivated incoherent operations [28, 64].

Physical Incoherent operation (PIO) : PIO is obtainedthrough a class of noncoherence generating operations on aprimary (A) and an ancillary system (B) [28, 64]. A gen-eral PIO operation consist of an unitary operation UAB on thestate ρA of system A and the incoherent state ρB of systemB, followed by a general incoherent projective measurementon system B. The PIO admits following Kraus decomposi-tion Ki =

∑j e

iθj |πi(j)〉〈j|Pi and their convex combina-tions. The πi are permutations and Pi is an complete set oforthogonal incoherent projectors[28]. Orthogonal incoherentprojectors are those which does not introduce any coherencein the system after measurement.

The PIOs are implementable using the aforementionedmethod and additionally it allows incoherent measurementsin environment and classical post-selection on the outcomes.

It is evident now that the MIO is the largest set of incoherentoperations, and others are strict subset of it. The nontrivialrelationship can be depicted in the following way [28, 61, 64,65]

PIO ⊂ SIO ⊂ IO ⊂MIO. (21)

A special subset of PIO is considered and discussed in [38].These are very important in the sense that they preserve co-herence of the input states.

Coherence preserving operation (CPO) : A channel Φis CPO iff it keeps the coherence of a state invariant, i.e.,C(Φ[ρ]) = C(ρ), where C is an arbitrary coherence mea-sure. The Kraus operator of CPO is expressed as K =∑i e

iθi |π(i)〉〈i|.We also consider the following two incoherent operations

introduced in Ref.[61].Genuinely Incoherent operation (GIO) : A channel Φ is

GIO iff Φ[δ] = δ, i.e., all incoherent states are fixed pointsfor the channel. Therefore, GIO does not allow transforma-tion between any incoherent states. All Kraus operators forthis operation are diagonal in the incoherent basis.

Fully Incoherent operation (FIO) : A quantum operation isfully incoherent if and only if all Kraus operators are incoher-ent and have the same form. Kraus operators are incoherentmeans, KiδK

†i is an incoherent state as well. This means that

only pure incoherent states are free in this resource theory.There exist other concepts of free operations in the context

of resource theory of coherence [3], but we limit ourselves tothe operations described above.

6

The following Kraus representations are the possible FIOsfor single qubits(

a1 b10 0

),

(a2 b20 0

);

(0 0a1 b1

),

(0 0a2 b2

);(

0 d1

c1 0

),

(0 d2

c2 0

);

(c1 00 d1

),

(c2 00 d2

);(22)

where |a1|2 + |b1|2 = 1 = |a2|2 + |b2|2 and a1b∗1 + a2b

∗2 =

0 = b1a∗1 + b2a

∗2, and |c1|2 + |c2|2 = 1 = |d1|2 + |d2|2. From

Eq. (22) one can easily check that all the matrices of a Krausrepresentations have the same form. As an example, the firsttwo matrices have nonzero entries only in the first row. Thelast one is the GIO for the qubit case. Note that first two FIOshave zero coherence. The coherence and purity of last twoFIOs are Cl1 = |d1c

∗1 + d2c

∗2| and P = 1

2 (1 + C2l1

). Hence,we have the relation 2P − C2

l1= 1 with P ∈ [ 1

2 , 1].According to the Ref. [66], any qubit incoherent operation

(IO) admits a decomposition with at most five Kraus opera-tors. A canonical choice of Kraus operators for IO is(

a1 b10 0

),

(0 0a2 b2

),

(a3 00 b3

),

(0 b4a4 0

),

(a5 00 0

),

(23)where one can choose ai ∈ R while bi ∈ C. Further,∑5i=1 a

2i =

∑4j=1 |bj |2 = 1 and a1b1 + a2b2 = 0 holds.

The coherence and purity of IOs are Cl1 =∑4i=1 ai|bi| and

P = 12 [1−µ(1−µ)−κ(1−κ)+

∑4i=1 a

2i |bi|2], respectively,

where µ = (a22 + a2

4) and κ = (|b1|2 + |b4|2).Similarly, Ref. [66] shows that the canonical set of Kraus

operators for SIO is(a1 00 b1

),

(0 b2a2 0

),

(a3 00 0

),

(0 0a4 0

), (24)

where ai ∈ R and∑4i=1 a

2i =

∑2j=1 |bj |2 = 1 holds. The

coherence and purity of SIOs are Cl1 = a1|b1| + a2|b2| andP = 1

2 [1−ν(1−ν)+ |b1|2|b2|2 +∑2i=1 a

2i |bi|2], respectively,

with ν = (a21 + a2

3).The CoPu diagrams in Fig.(3) show that SIOs are subset of

IOs. Note that all of the purity range is not allowed for bothSIOs and IOs.Observtion 3: It is possible to distinguish between SIO andIO for some regions of CoPu diagram in Fig. 3. Accord-ing to the Ref.[66, 67], if one considers state transformationby incoherent operations, qubit SIOs and IOs are equivalent.However, the above observation tells us the opposite behavior.

The Kraus representation of all possible single qubit PIOsare given by [67](

eiθ1 00 0

),

(0 00 eiθ2

);

(0 0eiφ2 0

),

(0 eiφ1

0 0

);(

eiθ1 00 0

),

(0 eiφ1

0 0

);

(0 0eiφ2 0

),

(0 00 eiθ2

);(25)(

eiθ1 00 eiθ2

);

(0 eiφ1

eiφ2 0

), (26)

Co

her

ence

0.2

0.4

0.6

0.8

1

Purity0.2 0.4 0.6 0.8 1

FIG. 3: (Color online) The allowed coherence-vs-purityregion for IO (region inside black curve) and SIO (region

inside red curve) respectively. Coherence of the channels ismeasured by l1-norm. The figure depicts the well known

phenomenon that SIO ⊂ IO. Moreover, channels outsidethe overlap region are IO and can be easily distinguished

from the SIO.

where first four PIOs are the coherence breaking channels andhave zero coherence in both KROC and AROC, and the lasttwo PIOs are the all possible single qubit CPOs and have unitcoherence and unit purity in KROC.

Although we have expected that the coherence of the inco-herent channels will be zero, it turns out to be not so. How-ever, we draw the following observation from the incoherentchannels considered in this section.Observation 4: All qubit incoherent channels which are ei-ther unital or nonunital, cannot create coherence in the sub-system ‘A’ of its Choi matrix. It can easily be verified fromthe Table.I.

Proof. Here we will try to prove the Observation.4 for IO, SIOand PIO. If we consider the Kraus decomposition of IO asgiven in Eq.(20), then its Choi matrix will be

ρAB =1

d

∑i

Ki ⊗ I

(∑lm

|ll〉〈mm|

)K†i ⊗ I,

=1

d

∑cijc∗it|fi(j)〉〈j|l〉〈m|t〉〈fi(t)| ⊗ |l〉〈m|,

=1

d

∑cijc∗it|fi(j)〉〈fi(t)| ⊗ |l〉〈m|δjlδmt,

=1

d

∑cilc∗it|fi(l)l〉〈fi(t)t|.

7

Now the reduced density matrix of the subsystem A is

ρA =1

d

∑cilc∗it|fi(l)〉〈fi(t)| ⊗ 〈n|l〉〈t|n〉,

=1

d

∑cilc∗il|fi(l)〉〈fi(l)|. (27)

Therefore, ρA is incoherent for IO. This also guarantees thatρA will be incoherent for SIO and PIO. Although we do nothave direct proof for other type of incoherent operations, thefollowing Table.I confirms that the Observation.4 is also trueatleast for single qubit FIO and GIO.

This observation says that the nonunital channels which has~τ = 0, 0, τz qualifies as potential candidates for incoherentoperations (see Table.I).

Coherence ofChannels ρAB ρA τz

IO [0, 1] 0 *SIO [0, 1] 0 *

PIO (CPO) 0 (1) 0 *FIO (GIO) #([0, 1]) 0 # (0)

TABLE I: Table shows that all qubit incoherent operationshave zero coherence in ρA(= TrA[ρAB ]). The ∗ denotes thatthe corresponding channels are in general nonunital. The #

for FIOs indicates that the channels which have zerocoherence (in Choi matrix) are nonunital otherwise they are

unital.

A. Coherence Non-Generating Channel (CNC)

A CPTP map, Φ which does not generate quantum coher-ence from an incoherent state is known as the coherence non-generating channel [62], i.e., Φ[I] ⊂ I. The incoherent op-erations are strict subset of these channels. These channelsare different from the set of incoherent operations in the sensethat the monotonicity of coherence may break under these op-erations while acting on one subsystem [62].Proposition 2: For general qubit CNC channels, 0 ≤ Cl1 ≤√

2.Proof. A full rank qubit channel is CNC iff it admits followingtwo Kraus decompositions [62]. The first one is

K1 =

(eiη cos θ cosφ 0

− sin θ sinφ eiξ cosφ

),

K2 =

(sin θ cosφ eiξ sinφ

e−iη cos θ sinφ 0

),

where θ, φ, ξ, η ∈ R. Notice that K1 and K2 may not in-dividually be incoherent but K1(·)K†1 + K2(·)K†2 can be if

sinφ cosφ sin θ cos θ = 0. Therefore, CNC channels may notbe incoherent.

The coherence and purity of the above channel are givenby Cl1 = cos θ + | sin θ sin 2φ| and P = 1

8 (5 + cos 2θ +

2 cos2 θ cos 4φ), respectively. The incoherent condition willalways guarantee that the coherence will be less than or equalto 1. Otherwise, the coherence of CNC can be 0 ≤ Cl1 ≤

√2.

The coherence will reach its maximum at θ = π4 = φ.

The other CNC channel is given by

K1 =

(cos θ 0

0 eiχ cosφ

)and K2 =

(0 sinφ

eiχ sin θ 0

).

This channel is an incoherent channel. The coherence and pu-rity of this CNC channel is Cl1 = cos θ cosφ + | sin θ sinφ|and P = 1

16 (10 + cos 4θ+ 4 cos 2θ cos 2φ+ cos 4φ), respec-tively and 0 ≤ Cl1 ≤ 1.

The Fig. (4) shows allowed range of all CNC channels. Itis clear that allowed region of incoherent CNCs is inside theregion of all CNCs. From the CoPu diagrams it is clear thatfor these channels, purity ranges from 1

2 to 1.

CMC

CNC

Co

her

ence

0

0.5

1

1.5

2

2.53

Purity

0.5 0.6 0.7 0.8 0.9 1

FIG. 4: (Color online) The allowed coherence-vs-purityregion for CMC and CNC. The region inside the red curve isfor CNC and if the CNCs are incoherent then its coherencelie in the region between black curves. The region betweenupper black line (y = 1 -line) and blue curves is for CMCs.

Note that for all these channels P ∈ [ 12 , 1].

V. COHERENCE OF OTHER KNOWN QUBIT CHANNELS

In this section we will consider some known qubit channelsand find its l1-norm coherence. We will investigate whetherthese channels can be characterized by its coherence and pu-rity.Channel to obtain maximum coherence (CMC): The max-imum value of the l1-norm coherence for two qubit system

8

is 3. This value is achieved by the state | + +〉, where|+〉 = 1√

2(|0〉+ |1〉). Hence, its obvious to search for a qubit

channel which will reach this value.Proposition 3: For general two qubit CMC channels, 1 6Cl1 6 3.Proof. The channel which may reach this value admits thefollowing Kraus decomposition

K1 =1√2

(cos θ1 e−iφ1 sin θ1

eiφ1 sin θ1 − cos θ1

),

K2 =1√2

(cos θ2 e−iφ2 sin θ2

eiφ2 sin θ2 − cos θ2

).

The coherence and purity of the channel is

Cl1 =1

4

(2 + ς +

∑j=±

gj + fj

),

P =1

16(11 + 3 cos 2θ1 cos 2θ2 + ς + `21 + `12), (28)

where g± = |e±2iφ1 sin2 θ1 + e±2iφ2 sin2 θ2|, f± =2|e±iφ1 sin 2θ1 + e±iφ2 sin 2θ2|, ς = cos 2θ1 + cos 2θ2 and`mn = 4 cosm(φ1 − φ1) sinm nθ1 sinm nθ2. The coherenceof the channel will reach its maximum, i.e., Cl1 = 3 forθ1 = π

4 = θ2 and φ1 = φ2. In fact, the coherence of thischannel obeys 1 ≤ Cl1 ≤ 3 which is confirmed by the CoPudiagram (see Fig. (4)). Therefore, CMC channel either in-creases or unalters the coherence of the state. Moreover, thischannel can be considered as coherence generating channel.

FIG. 5: (Color online) Plot of the l1-norm coherence ofdegradable (red regions) and anti-degradable channels (blue

regions) with the parameters θ and φ. It depicts that thewhole region is completely covered and degradable channels

lie in the range 1√2≤ Cl1 ≤ 1.

A family of qubit channels: A family of qubit channels can

be described by two Kraus operators in σz basis as [68]

K1 =

(cos θ 0

0 cosφ

)and K2 =

(0 sinφ

sin θ 0

), (29)

where θ, φ ∈ [0, π]. In AROC, the channel is described byλx = cos(φ−θ), λy = cos(φ+θ), λz = (cos 2φ+cos 2θ)/2,τx = τy = 0 and τz = (cos 2θ − cos 2φ)/2. The coherenceand purity of the channel are Cl1 = cos θ cosφ+ | sin θ sinφ|and P = 1

2 + 18 (cos 2φ+ cos 2θ)2.

We know that a CPTP map can be described as a unitarycoupling with the external environment. If a CPTP map Φchanges a state ρS to ρ′S , then it can be represented as

ρ′S = Φ(ρS) = TrE [USE(ρS ⊗ ωE)U†SE ], (30)

where USE is the unitary coupling between the system andthe environment E and ωE is a fixed state of E. In this pro-cess environment state also changes to ω′E . A CPTP map canalso be represented as operator sum representation as in Eq.(1). The final environment state can be found from the initialsystem state ρ by using the complementary channel Φ′ of Φas

ω′E = Φ′(ρS) = TrS [USE(ρS ⊗ ωE)U†SE ]. (31)

A map Φ is called degradable if there exits a third map Ω suchthat Φ′ = ΩΦ and Ω takes the state ρ′S to ω′E[68]. Similarly amap is antidegradable if there exist a Ω such that Φ = ΩΦ′ andwhich take the final environment state ω′E to ρ′S [68]. Moredetails can be found in the reference [68]. The channel rep-resented in Eq. (29) is degradable for cos 2θ

cos 2φ ≥ 0, otherwiseanti-degradable [68], see Fig.(5). The Fig.(5) shows that thecoherence of degradable channel satisfies 1√

2≤ Cl1 ≤ 1. The

CoPu diagram in Fig.(7) also confirms this observation.Observation 5: If the channel in Eq.(29) is anti-degradablethen its coherence be always less than 1√

2.

For cos 2θ = 1 and cos 2φ = 2η − 1, it describes the am-plitude damping (AD) channels with damping rate η. Thecoherence and purity of AD channels are Cl1 =

√η and

P = 12 (1 + C4

l1) respectively. Equivalently, Cl1 = 4

√2P − 1.

As it is a non-unital channel, the CoPu diagram for this chan-nel is shown in Fig.(7).

If sin θ = ± sinφ, the above channel becomes unital.Specifically, for θ = φ, the channel becomes a bit flip channelbut for θ = −φ, it is a bit-phase flip channel. The coher-ence and purity of both bit flip and bit-phase flip channels areCl1 = cos(2θ) and P = 1

4 (1 + 2C2l1

) respectively or equiv-alently, 2P − C2

l1= 1

2 , with P ∈ [ 14 ,

34 ]. The above channel

will unitarily transform to a phase flip channel (decoherencechannel) if we multiply the Kraus operators with Hadamardgate [68]. All the Pauli channels are unital channels. Thus,the CoPu diagrams of these channels can be depicted insidethe CoPu diagram of general unital channels (see Fig. (6)).Qubit Decoherence, Depolarization and Homogenizationchannels: The decoherence, depolarization and homogeniza-tion are nonunitary channels and form Markovian semigroup[69].

9

Co

her

ence

0.2

0.4

0.6

0.8

1

Purity

0.2 0.4 0.6 0.8 1

FIG. 6: (Color online) Figure depicts the coherence vs puritycurve for unital channels. The red curve depicts the

decoherence channels which coincides with the lowerboundary of CoPu curve of unital channels. The green curverepresents the depolarizing channels and the orange one is for

the bit flip as well as bit-phase flip channels.

The decoherence is a process in which the off-diagonalterms of the density matrix of a quantum system are contin-uously suppressed in time, i.e., ρ → ρt→∞ = diag(ρ). Thedecoherence channel is described by λx = λy = e−

tT and

λz = 1. This channel is a unital channel. The coherence andpurity of this channel are given by

Cl1 = e−tT and P =

1

2(1 + C2

l1) (32)

respectively. Now we have 2P − C2l1

= 1. As P ∈ [ 12 , 1],

the decoherence channels will represent the minimum coher-ence boundary of unital channels. The CoPu diagram for thischannel is in Fig.(6).Observation 6: For the qubit decoherence channels, the con-currence and the l1-norm coherence are same.The above observation can be easily verified as the concur-rence of the decoherence channel is e−

tT [69].

The depolarizing channel with noise parameter p transmitsan input qubit perfectly with probability 1− p and outputs thecompletely mixed state with probability p, i.e., ρ → ρf =(1 − p)ρ + pI/2. The depolarization channel is described asλx = λy = λz = e−

tT . This channel is also a unital channel.

The coherence and purity of this channel are

Cl1 = e−tT and P =

1

4(1 + 3C2

l1), (33)

respectively. Therefore, 4P−3C2l1

= 1, with P ∈ [ 14 , 1]. This

restriction is represented in CoPu diagram (see Fig.(6)).The homogenization is an evolution that transforms the

whole Bloch sphere into a single point, i.e., it is a contractive

map with the fixed point (the stationary state of the dynam-ics). This map is described by λx = λy = e−

tT2 , λz = e−

tT1 ,

τx = τy = 0 and τz = ω(1 − e−t

T1 ), where the parameters,ω is the purity of the final state, T1 is the decay time, T2 is thedecoherence time. It is a non-unital process. The coherenceand the purity of this channel are

Cl1 = e−t

T2 andP =1

4[1+e−

2tT1 +2e−

2tT2 +ω2(1−e−

tT1 )2],

(34)respectively. For ω = 1 and T2 = 2T1, we have the re-lation between the coherence and purity as given by Cl1 =4√

2P − 1. Some CoPu diagrams of this channel are shown inFig. (7).

Co

her

ence

0

0.25

0.5

0.75

1

1.251.5

Purity0.2 0.4 0.6 0.8 1

FIG. 7: (Color online) Figure depicts the coherence vs puritycurve for non-unital channels. The green region represents

the anti-degradable channels whereas the yellow regionshows the degradable ones. The lower boundary of these twochannels coincides with the red curve which represents theamplitude damping channels. The blue, orange, red and the

purple curve represent the homogenization channels forT2 = T1, 2T2 = T1, T2 = 2T1 and T2 = 5T1 respectively

with ω = 1. Note that the T2 = 2T1 curve coincides with ADcurve.

VI. RELATIVE ENTROPY OF COHERENCE ANDHOLEVO QUANTITY

Here, we find a complimentarity relation between the rela-tive entropy of coherence and the Holevo quantity of the chan-nel. Recently, there has been an attempt to find such relationinvolving quantum coherence and the information process-ing quantities like superdense coding capacity, teleportationfidelity etc [70]. Infact, in Ref. [71], authors try to relate the

10

Holevo quantity with the loss of coherence due to the pro-jective measurement in one of the subsystems of an arbitrarybipartite density matrix.

The Holevo quantity is lower bound to the classical capacityof the quantum channel. In fact, the Holevo quantity is exactlyequal to the classical capacity for all entanglement breakingchannels, depolarizing channels and qubit unital channels. Letus define the Holevo quantity, χ. Let Alice encode the in-formation of an random variable X in an quantum ensem-ble px, ρx and sends it to Bob through a quantum channel.In order to extract the information about X , Bob will per-form positive operator valued measurement (POVM) on thereceived state and record the outcomes which is another ran-dom variable Y . Then we have the following inequality

I(X : Y ) ≤ S(∑x

pxρx)−∑x

pxS(ρx), (35)

where S(ρX) −∑x pxS(ρx) = χ, the Holevo quantity [54].

It gives the upper bound to the accessible information aboutX by Bob.

The classical correlations [72] and the Holevo quantity arerelated concepts [73]. A bipartite state will have classical cor-relations if after application of rank one POVM (i.e., entangle-ment breaking operation) on one of the parties will transformthe state to either classical-quantum (CQ) or quantum clas-sical (QC) states [73]. The classical correlations [72] of aquantum state ρSA is given by,

J = supΠj

S(ρS)−∑j

pjS(ρS|ΠAj

)

, (36)

where ρS|ΠAj

are the post measurement states with probabilitypi due to the application of projective measurements (Πj)on the part A of the Choi matrix ρSA. Application of projec-tive measurements on the A, the average post measurementstate will be of the form of QC state, i.e.,

ρQC =∑i

piρiQ ⊗ |i〉〈i|C , (37)

where ρi are not orthogonal. Therefore, the classical correla-tions of the state ρSA should be equivalent to the maximumpossible mutual information of ρQC , i.e., J = I(ρQC) =S(∑i piρi)−

∑i piS(ρi), therefore, the classical correlations

is nothing but the Holevo quantity. Hence, the Holevo quan-tity of a channel is equal to the classical correlations of itsChoi matrix [73].

The classical correlation, i.e., the Holevo quantity isbounded by the relation χ(Φ) 6 log d. This observation leadsus to derive a complementarity between the relative entropyof coherence and the Holevo quantity of a channel.

Theorem – The complementarity relation between theHolevo quantity and the channel coherence is given by

Cr(Φ) + χ(Φ) 6 2 log d, (38)

where d is the dimension of the system B(A).

Proof. Under state-channel duality, we have

ρAB = (Φ⊗ I)|Ψ〉AB〈Ψ| ≡ Φ(|Ψ〉AB). (39)

The relative entropy of coherence for ρAB is defined in thebasis |µi〉|i〉, where |µ〉 is eigen basis of ρA and |i〉is the basis in which a rank one projective measurement isperformed on B. Therefore, we have

Cr(ρAB) = S(ρDAB)− S(ρAB). (40)

The Holevo quantity or the classical correlation is given by

χ(Φ) = J(ρAB) = S(ρA) + S(ρDB )− S(ρDAB), (41)

where S(ρDAB) =∑i(I ⊗ Πi)ρAB(I ⊗ Πi) with Πi = |i〉〈i|.

Thus we have

Cr(ρAB) + χ(Φ) = S(ρA) + S(ρDB )− S(ρAB). (42)

Using the triangle inequality for S(ρAB), i.e., S(ρAB) ≥S(ρA)−S(ρB), and the fact that S(ρB) ≤ log d and S(ρDB ) ≤log d, we have

Cr(ρAB) + χ(Φ) ≤ 2 log d. (43)

Hence the proof.

The above relation shows that the more coherence a channelhas, the less will be its Holevo quantity.

VII. DISCUSSION AND CONCLUSION

The significance of this work is three fold: Choi-Jamiołkowski isomorphism allows us to associate a densitymatrix with a channel. The purity and coherence of this den-sity matrix can be fruitfully associated with the channel. Us-ing CoPu diagrams we have shown that it may be possible todistinguish different qubit channels. Distinguishing the chan-nels using CoPu diagrams depicts the inter-relation betweenpurity and the coherence of the channels. It has a broadermeaning also, e.g., given a purity one may not find a channelwhich has certain amount of coherence. These relations be-tween coherence and purity show deeper restriction on avail-able coherent channel.

If we look closely, we find that the purity and coherencesatisfy the following equation in some of the cases, i.e.,

$P − ϕC2l1 = 1, (44)

where $,ϕ ∈ R.We can rewrite this as,(√

$

ϕ

√P)2

− (Cl1)2 =

(1√ϕ

)2

. (45)

This expression can have an interesting interpretation. Wecan think of scaled

√P as time component, and coherence

as the spatial component of a vector in two-dimensionalMinkowski space. This vector is timelike.

11

Boundaries of light cone are given by the following relation(Cl1√P

)2

=$

ϕ. (46)

In general, Cl1√P ≤ ±

√$ϕ . Physically, this means that the

allowed regions in CoPu diagrams are restricted by the aboverelation. The values are further constrained by (45) and lieinside hyperbolas within the above light cone boundaries.

Secondly, the CoPu diagram can help us in distinguishingdifferent qubit channels, eg., the unital and nonunital, the in-coherent channels, degradable and antidegradable, Pauli chan-nels, etc. These studies unveil very interesting properties ofthese channels. For example, we find that the qubit incoherentchannels can either be unital or nonunital with ~τ = 0, 0, τz.We also find that all coherence breaking channels has zero co-herence. However, this is not usually true for entanglement

breaking channels. We observe that the coherence preservingqubit channels have unit coherence.

Thirdly, we find a complimentarity relation between rela-tive entropy of coherence and the Holevo quantity of the chan-nel. It says that if channel has more coherence, the Holevoquantity of the channel will be restricted. Naturally, this rela-tion is very much important in quantum information process-ing tasks.

Although we mainly focus our study for single qubits, itwill be interesting to extend our results for higher dimensions.There are many indications in our work that says one mightbe able to do it. We hope our findings will open up a newdirection of studies towards the resource theory of coherence.

Note: We notice a work [74] appeared in the arXiv on thesame day of submission of our work in the arXiv. Althoughtheir analysis is very different than ours, the definition of co-herence for quantum channels has overlap with our method.

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