entanglement or separability - zentraler...

Bachelor Thesis

Entanglement or Separabilityan introduction

Lukas Schneiderbauer

December 22, 2012

Quantum entanglement is a huge and active research field these days. Not only thephilosophical aspects of these ’spooky’ features in quantum mechanics are quite interesting,but also the possibilities to make use of it in our everyday life is thrilling. In the last fewyears many possible applications, mostly within the ’Quantum Information’ field, have beendeveloped.Of course to make use of this feature one demands tools to control entanglement in

a certain sense. How can one define entanglement? How can one identify an entangledquantum system? Can entanglement be measured? These are questions one desires ananswer for and indeed many answers have been found.However today entanglement is not yet fully in control by mathematics; many problems

are still not solved. This paper aims to provide a theoretical introduction to get a feelingfor the mathematical problems concerning entanglement and presents approaches to handleentanglement identification or entanglement measures for simple cases.The reader should be aware of the fact that this paper constitutes in no way the claim

to be a summary of all available methods, there exist many more than demonstrated in thefollowing pages.

Student ID number 0907633Degree course Physics

assisted by ao. Univ.-Prof. i.R. Dr. Reinhold Bertlmann

Contents

1 Introduction 3

2 Preliminary definitions 42.1 Composite quantum systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2 Density operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.2.1 Reduced density operator for a bipartite quantum system . . . . . . . . . . . . 42.3 Entangled states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.3.1 A pure correlated composite state . . . . . . . . . . . . . . . . . . . . . . . . . . 52.4 Hilbert-Schmidt space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.5 Qudit systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.5.1 Qubit systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

3 Pure bipartite qudit states 73.1 Schmidt decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.2 Von-Neumann entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

4 Mixed bipartite qudit states 114.1 A criterion for non-entanglement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114.2 Generalization of the Von-Neumann entropy . . . . . . . . . . . . . . . . . . . . . . . . 12

4.2.1 Requirements for entanglement measures . . . . . . . . . . . . . . . . . . . . . . 124.2.2 Entanglement measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

4.3 Generalization of the Schmidt rank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144.4 Entanglement witnesses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

4.4.1 Entanglement Witness Theorem (EWT) . . . . . . . . . . . . . . . . . . . . . . 154.4.2 Positive Map Theorem (PMT) . . . . . . . . . . . . . . . . . . . . . . . . . . . 154.4.3 Positive Partial Transpose (PPT) Criterion . . . . . . . . . . . . . . . . . . . . 164.4.4 Bertlmann-Narnhofer-Thirring Theorem . . . . . . . . . . . . . . . . . . . . . . 17

5 Multipartite states 205.1 Some entanglement measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

5.1.1 Geometric measure of entanglement . . . . . . . . . . . . . . . . . . . . . . . . 205.1.2 Measure of entanglement by Barnum . . . . . . . . . . . . . . . . . . . . . . . . 20

5.2 An entanglement witness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

6 The choice of factorization 226.1 The choice of factorization for pure states . . . . . . . . . . . . . . . . . . . . . . . . . 226.2 The choice of factorization for mixed states . . . . . . . . . . . . . . . . . . . . . . . . 22

7 Conclusion 24

References 24

1 INTRODUCTION

1 Introduction

Quantum entanglement, named by Erwin Schrödinger1, is a quantum mechanics phenomenon whichwas first outlined by Albert Einstein, Boris Podolski and Nathan Rosen in 1935 and led to the fa-mous EPR paradoxon.[6] Their result was that quantum mechanics can’t be a correct theory sinceentanglement would lead to a violation of the classical principle of local realism:

»We are thus forced to conclude that the quantum-mechanical description of physicalreality given by wave functions is not complete.« - Einstein, Podolski, Rosen [6]

On the other hand quantum mechanics was succussfully confirmed by experiments. Therefore theidea of ’hidden variables’ arose, meaning quantum theory is not fundamental but a statistical theorywhich covers the fundamental theory. This makes it possible to deny a violation of local realism asmatter of principle and at the same time to ’believe’ in the success of the theory.1964 John Stewart Bell provided a way to determine experimentally whether hidden variables exist

or not (see his Bell inequalities[4]). Surprisingly the experiments turned out to deny the existence ofsuch hidden variables.Although the interpretation of quantum theory may vary, today quantum mechanics is widely ac-

cepted as fundamental theory by physicists.

This work is an introduction developing criteria to identify entangled quantum systems for specificcases. To start at an uniform level it first provides some fundamental mathematical definitions insection 2. Mathematical entities with physical meaning like the Hilbert space or a density operatorare defined. Also a (mathematical) answer to the question »What is entanglement?« is given. Thefollowing derivations are based on these definitions.Next it faces one of the simplest non trivial problems in section 3: a pure bipartite qudit state. Two

important concepts, the Schmidt-decomposition and the Von-Neumann entropy, are introduced whichwill prove to be useful in further studies. The Bell states will function as examples.In chapter 4 mixed bipartite qudit systems are discussed. First a simple but important criterion for

non-entanglement is derived. Generalization approaches of the Von-Neumann entropy and the Schmidtrank are presented as entanglement measures which will include the entanglement of formation, therelative entropy of entanglement, the entanglement of distillation, entanglement cost and a Hilbert-Schmidt measure. Also the concept of entanglement witnesses with a few applications like the PositivePartial Transpose Criterion is displayed.Furthermore a small outlook for multipartite systems is given in section 5. Two more measures of

entanglement are presented and a specific entanglement witness for an N -partite system is given.And last but not least the issue of the possibility to choose different algebra factorizations with

respect to the consequences to entanglement is discussed in chapter 6.

1The original German name for this ’spooky’ feature was »Verschränkung«.

3

2 PRELIMINARY DEFINITIONS

2 Preliminary definitions

2.1 Composite quantum systems

A quantum system is represented by a Hilbert space H. Let’s consider a number of such systems,denoted by HA, HB and so forth.

Definition 2.1. It is postulated that the composite system of these subsystems HA, HB, ... isrepresented by their Product-Hilbert space HAB...

HAB... = HA ⊗HB ⊗ ... (2.1)

An operatorO in a composite system SAB... is denoted byOAB... . For a product state |ΨA〉⊗|ΨB〉⊗...one also writes |ΨA,ΨB, ...〉.

2.2 Density operator

Definition 2.2. Given an ensemble {|ϕi〉, pi} of N possible pure states |ϕi〉 with probability pi onedefines the density operator ρ

ρ :=

N∑i=1

pi|ϕi〉〈ϕi|,N∑i=1

pi = 1 (2.2)

Note that the |ϕi〉 are not the eigenstates of ρ, therefore not orthogonal in general. This densityoperator ρ represents a mixed state2 (see e.g. [2]) of a quantum system and has the following importantproperties:

〈ϕ|ρ|ϕ〉 ≥ 0 ∀|ϕ〉 ∈ H ⇐⇒ ρ† = ρ (2.3)

tr(ρ) = 1 (2.4)

Proof. Eq. (2.3) is obvious and eq. (2.4) can be shown by just using the definition of the trace:tr(ρ) ≡

∑i〈Ψi|ρ|Ψi〉 with an arbitrary ON basis {Ψi}.

2.2.1 Reduced density operator for a bipartite quantum system

Definition 2.3. Let ρAB be a density operator on HAB. Then the reduced density operator ρA isdefined as

ρA := trB(ρAB), (2.5)

whereas trB(ZAB) is called a partial trace, which is defined by trB(ZAB) :=∑

n〈ΨBn |ZAB|ΨB

n 〉 witharbitrary basis {|Ψi〉}. The result is an operator on HA. More to partial traces can be found in [2].The reduced density operator ρA can be envisioned as the state in the subsystem SA. All probability

predictions for local measurements (the related observable is in the form of A⊗ I) on system SA canbe allocated to the reduced density operator.

2A mixed state is a generalization of a pure state. To see this, set N = 1 in eq. (2.2) and one gets the density operatorof a pure state, namely ρpure = |ϕ〉〈ϕ|.

4

2.3 Entangled states 2 PRELIMINARY DEFINITIONS

2.3 Entangled states

Definition 2.4. A composite state is called correlated if and only if its density operator ρAB... cannot be written as a product operator, i.e.

ρAB... 6= ρA ⊗ ρB ⊗ ... (2.6)

Since ρ represents a mixed state in general, correlation alone may not necessarily imply a deviationfrom classical views. To classify a non classical effect we go on with further definitions.

Definition 2.5. A state is called separable if and only if the density operator ρAB... can be written as

ρAB... =

n∑r=1

pr ρAr ⊗ ρBr ⊗ ... (2.7)

Note, that for n = 1, ρAB... is not correlated. The state for n 6= 1, that is a separable correlatedstate, is called a classical correlated state. From the definition it is clear that the family of separablestates is a convex set3.

Definition 2.6. A state is called entangled if and only if the state is correlated and not separable, i.e.

ρAB... 6=n∑r=1

pr ρAr ⊗ ρBr ⊗ ... (2.8)

It may be useful to take a look at a special case: a pure and correlated state.

2.3.1 A pure correlated composite state

Claim. A pure correlated composite state is an entangled one.

Proof. Let ρ ≡ ρAB... be the density operator of a pure state |Ψ〉 ≡ |ΨAB...〉, i.e. ρ = |Ψ〉〈Ψ|. Choosea vector |Φ〉, so that 〈Ψ|Φ〉 = 0.First, we make the attempt to decompose ρ to a convex sum4 of other density operators:

ρ =

n∑r=1

λrρr

=⇒ 〈Φ|ρ|Φ〉 = 0 =n∑r=1

λr〈Φ|ρr|Φ〉

Since all 〈Φ|ρr|Φ〉 are positive (see eq. (2.3)) and λr are positive, the above equation holds, if andonly if 〈Φ|ρr|Φ〉 = 0.Now complete |Ψ〉 to an orthonormal basis {|φk〉}, |φ1〉 ≡ |Ψ〉 and look at the matrix elements

of ρ and ρr in this basis: 〈φi|ρr|φj〉. We conclude, that all elements vanish, except for 〈Ψ|ρ|Ψ〉 =〈Ψ|ρr|Ψ〉 = 1, because of eq. (2.4). Therefore ρ = ρr ∀r ∈ [1;n]. Thus a decomposition is not possible.Furthermore let’s assume, ρ is correlated, i.e. ρ 6= ρ1⊗ ρ2⊗ ..., then ρ cannot be written in the form

of eq. (2.7). Hence it must be entangled.

That implies:|ΨAB...〉 6= |ΦA〉 ⊗ |ΦB〉 ⊗ ... ⇐⇒ |ΨAB...〉 entangled (2.9)

3A set C is said to be convex if (1− t)x+ t y ∈ C, ∀(x, y) ∈ C, ∀t ∈ [0,1].4The convex sum of ρ is ρ =

∑nr=1 λrρr,

∑nr=1 λr = 1, λr > 0

5

2.4 Hilbert-Schmidt space 2 PRELIMINARY DEFINITIONS

2.4 Hilbert-Schmidt space

Definition 2.7. Let H be a Hilbert space describing a quantum mechanical system and H the setof operators acting on H. Then the set H forms a Hilbert space for itself. This space is called aHilbert-Schmidt space. The scalar product on A denoted by 〈B,C〉 (B,C ∈ H) is defined by

〈B,C〉 := tr(B†C) (2.10)

and induces the Hilbert Schmidt norm ||B|| :=√〈B,B〉 .

2.5 Qudit systems

Definition 2.8. A quantum system with d linear independent states is called Qudit system and it isdescribed in a d-dimensional Hilbert space Hd.

2.5.1 Qubit systems

A qubit system is an important specific case of a qudit system, namely a system with d = 2. Thesystem is represented by a 2-dimensional Hilbert space H2. To name a few physical examples:

• Spin-12 -particle

• Polarization of single photons

• Quantum dots

Bell states

It is not the purpose to discuss Bell states in detail here, however, it makes a good example for anentangled bipartite qubit state as we will see later on.

Definition 2.9. Let |0〉 and |1〉 be the eigenstates of the Pauli operator5 σ3 acting on H2, then wedefine the so called Bell basis also known as Bell states: |Φ±〉, |ψ±〉 ∈ HAB = HA2 ⊗HB2 .

|Φ±〉 :=1√2

(|0, 0〉 ± |1, 1〉), |ψ±〉 :=1√2

(|0, 1〉 ± |1, 0〉) (2.11)

One could calculate its reduced density operators (see eq. (2.5)) which are in the form:

ρA = ρB =1

2I (2.12)

5Since the Pauli operators are not the subjects here, a detailed description is omitted, but can be referred in e.g. [2].

6

3 PURE BIPARTITE QUDIT STATES

3 Pure bipartite qudit states

After the basic but important definitions we will start discussing pure bipartite systems. A veryimportant concept is the so called Schmidt decomposition which can be applied to all pure bipartitestates, sadly only for those.In preparation for more complicated matters, e.g. mixed states, the Von-Neumann entropy and its

properties will be discussed.

3.1 Schmidt decomposition

For a pure bipartite qudit state |ΨAB〉 ∈ Hd it is useful to introduce a decomposition named theSchmidt decomposition, bi-orthogonal expansion or polar expansion. (see [7])

Theorem 3.1. Let |ΨAB〉 be a normed state in the composite system SAB in the Product-Hilbert-SpaceHAB = HA ⊗ HB with dimension dimHA = a and dimHB = b . The reduced density operators ofthe subsystems SA and SB are given by ρA = trB(ρAB) and ρB = trA(ρAB) with the density operatorρAB = |ΨAB〉〈ΨAB|.Then |ΨAB〉 can be written in the form (3.1) with normed eigenstates |uAn 〉, |wBn 〉 and eigenvalues

pAn = pBn of ρA and ρB, i.e.

ρA|uAn 〉 = pn|uAn 〉, pAn ∈ RρB|wBn 〉 = pn|wBn 〉, pBn ∈ R

|ΨAB〉 =

k∑n=1

√pn|uAn , wBn 〉, pn > 0 (3.1)

Proof. First, expand |ΨAB〉 into the basis of eigenstates of ρA and ρB:

|ΨAB〉 =

a,b∑i,j

〈uAi , wBj |ΨAB〉︸︷︷︸=:cij

|uAi , wBj 〉 (3.2)

The reduced density operator ρA can be written as a decomposition of his eigenstates:

ρA =

a∑n=1

pn|uAn 〉〈uAn |, pn

{> 0 0 ≤ n ≤ k= 0 k + 1 ≤ n ≤ a

,

a∑n=1

pn = 1 (3.3)

On the other side ρA must fulfill its role as a reduced density operator:

ρA = trB(ρAB) = trB(|ΨAB〉〈ΨAB|) 3.2=

a,b∑i,j

a,b∑k,l

cijckl trB(

|uAi 〉〈uAk |⊗|wBj 〉〈wBl |︷︸︸︷

|uAi , wBj 〉〈uAk , wBl |)

=

a,b∑i,j

a,b∑k,l

b∑m

cijckl〈wBm|wBj 〉〈wBl |wBm〉|uAi 〉〈uAk |

=

a,b∑i,j

a,b∑k,l

cijckl 〈wBl |wBj 〉︸︷︷︸δij

|uAi 〉〈uAk | =a,b∑i,j

a∑k

cijckj |uAi 〉〈uAk | (3.4)

By comparison of eq. (3.3) and (3.4) one finds that cmn = 0 for m 6= n and c2nn = pAn . After the

very similar calculation for ρB, one gets c2nn = pBn . Thus pAn = pBn ≡ pn. With these results it can

immediately be seen that eq. (3.2) reduces to eq. (3.1).

7

3.2 Von-Neumann entropy 3 PURE BIPARTITE QUDIT STATES

Definition 3.1. The number k in eq. (3.1) is called the Schmidt rank of |ΨAB〉, the eigenvalues {pn}are called Schmidt coefficients.

Claim. |ΨAB〉 is not entangled if and only if the Schmidt rank k = 1.

Proof. We know from eq. (2.9), that a pure state is entangled, if and only if the state is correlated.Since {|uAn 〉} and {|wBn 〉 form a basis, |ΨAB〉 can only be written in the form (2.9) if k = 1.

Claim. k = 1 ⇐⇒ tr((ρA)2) = tr((ρB)2) = 1.

Proof. k = 1 means, all eigenvalues {pn} of ρA = ρB ≡ ρ∗ vanish except of one. The trace of ρ∗ is 1(see eq. (2.4)). Now suppose k > 1. We know, pn > 0. Then ∀pn : pn < 1 =⇒ p2

n < pn. Thereforethe trace tr((ρ∗)2) =

∑kn p

2n < 1. Hence the trace tr((ρ∗)2) = 1 ⇐⇒ k = 1.

Let us summarize this important and useful statement: For a pure bipartite qudit state one canrecognize entanglement by evaluating the trace of the squared reduced density operators.What we have found is a nice criterion to detect entanglement.Another interesting side product of the Schmidt decomposition is the gained knowledge, that a

subsystem of a pure entangled state cannot be pure (remember that a pure state means rk(ρA) = 1and this is equal to k = 1).

Example. Schmidt decomposition and Bell statesUsing the above statement and the results from eq. (2.12) we can directly calculate the trace of the

reduced Bell states:

tr((ρA)2) = tr(1

4I) =

1

2

This finding is indeed interesting. It tells us, that the Bell states must be entangled.

3.2 Von-Neumann entropy

Another important tool to get hints about existent entanglement is the Von-Neumann entropy.[1]

Definition 3.2. The Von-Neumann entropy S(ρ) is defined as

S(ρ) := −tr(ρ log ρ) (3.5)

One can think of it as the quantum version of the entropy S known in thermodynamics. To makethis clear, one can for example show that the Von-Neumann entropy of the density matrices of themicro canonical, the canonical and the grand canonical ensemble coincides with the entropy S. As thethermodynamic entropy, the Von-Neumann entropy is a measure of information.For practical reasons it is often useful to express S in the eigenvalues of ρ:

Claim. The Von-Neumann entropy S(ρ) can be written in the form

S(ρ) = −d∑i=1

(λi log λi)

d∑i=1

λi = 1 (3.6)

with the eigenvalues λi of ρ.

8


Proof. Evaluate the trace with respect to eigenbasis {ψi} of ρ:

S(ρ) = −tr(ρ log ρ) = −tr(log ρ ρ)

= −d∑i=1

〈ψi| log ρ ρ|ψi〉 = −d∑i=1

λi〈ψi| log ρ|ψi〉

= −d∑i=1

λi log λi 〈ψi|ψi〉︸︷︷︸=1

= −d∑i=1

λi log λi

which is the claimed formula.

The Von-Neumann entropy S(ρ) has among others the following properties:

1. The entropy of a pure state is its minimum value.

ρ = |Ψ〉〈Ψ| ⇐⇒ S(ρ) = 0 (3.7)

2. The entropy of a density operator with rank d fulfills

0 ≤ S(ρ) ≤ log d (3.8)

Proof. To see the first property, one recognizes that ρ has only one eigenvalue λ1 = 1 if and only if ρis a pure state. With this finding one can evaluate eq. (3.6), which tells S(ρ|pure) = 0.Property 2 can be proven by calculating extremal values of function (3.6). We need to include the

constraint∑

i λi = 1 in the discussion, therefore we rewrite the expression as

d∑i=1

λi = 1 =d−1∑i=1

λi + λd =⇒ λd(λi 6=d) = 1−d−1∑i=1

λi

and thus the entropy reads

S(λl 6=d) = −d−1∑i=1

λi log λi − pd(λl 6=d) log pd(λl 6=d)

With ∂λd∂λl

= −1 the derivate of S(λl 6=d) emerges as

∂S

∂λl= −

d−1∑i=1

(∂λi∂λl

log λi +∂λi∂λl

)− ∂λd∂λl

log λd −∂λd∂λl

= − log λl − 1 + log λd + 1 = − log λl + log λd

This is zero if λmaxl = λd =⇒ λmax

l = 1d , hence the local maximum Smax is determined.

Smax = S(1

d) = −

d∑i=1

1

dlog

1

d= log d

Since Smax is the only local maximum (on the whole domain including the boundaries) it is a globalone.The global minimum value Smin = 0 is obvious (all 0 ≤ λi ≤ 1).

9


We already know from section 2.3.1, that a pure state can only possess non classical correlations (ornone at all). Let’s take a look at the valuable example of a pure qubit state.

Example. Von-Neumann entropy and Bell statesLet’s calculate the entropy for our example, the Bell states ρAB and its reduced operator ρA = ρB.To find S(ρAB), we can use property (3.6) of the entropy S; since ρAB is pure per definition:

S(ρAB) = 0

This result shows no new findings, it just confirms the obvious: The pure Bell state is a state withmaximum information (as every pure state is of course).The reduced density operator ρA apparently possesses the eigenvalues {1

2 ,12}. We use eq. (3.6) and

get the entropy:S(ρA) = log 2 = log d

With property (3.8) one notices that this is the maximum value of the entropy. As a consequence, thesub state ρA is a maximal undetermined state.

With these considerations on an information theory level, one is tempted to define a measure ofentanglement:

Definition 3.3. Consider a pure state |ΨAB〉 with the corresponding density matrix ρAB and reduceddensity matrices ρA and ρB. Then define the entropy of entanglement E(ρAB) = E(Ψ)

0 ≤ E(Ψ) := S(ρA) = S(ρB) ≤ 1 (3.9)

and call it a measure of entanglement of the state |ΨAB〉. A state with E(Ψ) = log d, d = dimH iscalled maximal entangled (see the Bell state).

This is the first step to the more general approach of entanglement measures as we will see in chapter4. Such measures also play a very important role for mixed states.

10

4 MIXED BIPARTITE QUDIT STATES

4 Mixed bipartite qudit states

As mixed states describe a more complicate matter than pure states, there are less concrete criteriaavailable.This chapter begins with a derivation of a specific criterion for non-entanglement. Then entanglement

measures are introduced as a generalization of the Von-Neumann entropy. After a renewed discussionabout the Schmidt rank we will proceed with establishing a more geometrical approach by the so calledentanglement witnesses.

4.1 A criterion for non-entanglement

In section 3.1 we have seen that a sub-state6 cannot be pure if its pure super-state is entangled. Wewill generalize this now.[2]

Theorem 4.1. Let SAB be a bipartite qudit system described in HAB = HAd ⊗ HBe and let ρAB be amixed density operator acting on HAB in this system. The corresponding reduced density operator ofthe subsystem SA is ρA.Furthermore if ρA is pure then the state ρAB is separable (⇐⇒ not entangled).

Proof. The mixed density operator ρAB can be written in its spectral basis:

ρAB =∑n

sn|ΨABn 〉〈ΨAB

n |,∑n

sn = 1, sn > 0 (4.1)

For simplicity set |ΨABn 〉 ≡ |Ψn〉. Now expand |Ψn〉 to the spectral basis of ρA and ρB: {ui} and

{wj}: (cp. eq. (3.2))|Ψn〉 =

∑i,j

cn,ij |ui, wj〉 (4.2)

Using the expansion from eq. (4.2), eq. (4.1) reads

ρAB =∑n

∑i,j

∑k,l

sncn,ijcn,kl |ui, wj〉〈uk, wl|︸︷︷︸|ui〉〈uk|⊗|wj〉〈wl|

(4.3)

To find some constraints for the coefficients cn,ij , we evaluate the reduced density operator ρA in anarbitrary ON basis {|ϕr〉}:

ρA = trB(ρAB)

=∑n

∑i,j

∑k,l

sncn,ijcn,kl

=〈wl|wj〉=δij︷︸︸︷∑r

〈ϕr|wj〉〈wl|ϕr〉 |ui〉〈uk|

=∑n

∑i,j

∑k

sncn,ijcn,kj |ui〉〈uk| (4.4)

Because {|ui〉} is the eigenbasis of ρA, it can also be written as:

ρA =∑n

rn|un〉〈un| (4.5)

6It may be a slight abuse of the used vocabulary, nevertheless it’s a good aberration of “state of a subsystem”.

11

4.2 Generalization of the Von-Neumann entropy 4 MIXED BIPARTITE QUDIT STATES

By comparison of eq. (4.4) and eq. (4.5), cn,ij = 0 for i 6= j. Suppose now, that ρA is a pure state,i.e. |ρA〉 = |u1〉〈u1|. Hence all cn,ij are zero except cn,11. This implies that eq. (4.3) reduces to

ρAB =∑n

snc2n,11 (|u1〉〈u1| ⊗ |w1〉〈w1|) (4.6)

which is the form of a separable state (see eq. (2.7)).

To sum up, this means: If a system is in a pure state it cannot be entangled with anothersystem.

4.2 Generalization of the Von-Neumann entropy

We have seen in section 3.2 that for pure bipartite states the Von-Neumann entropy is a measure ofentanglement. For mixed states this measure fails to distinguish classical and quantum correlations.There are however some attempts to generalize the Von-Neumann entropy. The so called entropymeasures are designed to coincide on the special case of pure bipartite states and to be equal to theentropy of the reduced density operators.What succeeds now are the requirements for a general entanglement measure[5] followed by defini-

tions of some specific cases.[5],[16]Note that not all of these provide a rule to do a practical evaluation. Generally the calculation might

not even be possible.

4.2.1 Requirements for entanglement measures

The following properties for a “good” entanglement measure E are of course not mandatory and thediscussion for such requirements is still open. Actually some of the measures introduced below do notfulfill all of the following properties.

1. ρ is separable ⇐⇒ E(ρ) = 0

2. Normalization0 ≤ E(ρ) ≤ log d (4.7)

for an entangled state of a d-dimensional system.

3. LOCC (local operations and classical communication) cannot increase E.7

4. ContinuityE(ρ)− E(σ)→ 0 for ||ρ− σ|| → 0 (4.8)

5. Additivity : n identical copies of the state ρ should contain n times the entanglement of one copy.8

E(ρ⊗n) = nE(ρ) (4.9)

6. Subadditivity:E(ρ⊗ σ) ≤ E(ρ) + E(σ) (4.10)

7. Convexity : The entanglement measure should be a convex function, i.e.

E(λρ+ (1− λ)σ) ≤ λE(ρ) + (1− λ)E(σ) (4.11)

with 0 < λ < 1.7A measure which decreases under LOCC is called entanglement monotone.8A⊗m := A⊗A⊗ ...⊗A︸︷︷︸

m times

12

4.2 Generalization of the Von-Neumann entropy 4 MIXED BIPARTITE QUDIT STATES

4.2.2 Entanglement measures

Entanglement of formationDefinition 4.1. Let ρAB =

∑i piρi, ρi ≡ |ϕi〉〈ϕi| be a mixed bipartite state, then the i-th entropy

of entanglement Ei ≡ E(ρi) for the pure states ρi is given by eq. (3.9), i.e. E(ρi) = S(trB(ρi)). Theentanglement of formation also known as entanglement of creation is then defined as

EF (ρAB) := min{dec}

∑i

piEi (4.12)

where min{dec}

means the minimum over all possible decompositions of ρAB.

To express this formula in words: Form the average entanglement entropy for pure states over adecomposition. Do this with all possible decompositions and pick the minimum.

Relative entropy of entanglementDefinition 4.2. Let ρAB be a mixed bipartite state (see eq. (2.7)) and σAB a separable one in thesystem SAB. Then one defines the relative entropy of entanglement ER(ρAB).

ER(ρAB) := minσAB∈SAB

tr[ρAB(log ρAB − log σAB)] (4.13)

To understand the structure of this formula let’s compare it with the original entropy S(ρ) =−tr(ρ log ρ). ER looks like “tr(ρ log ρ

σ )”9. So one could speak of this as the entropy of a state relativeto an arbitrary separable one. As above, form this entropy for all available separable states and takethe minimum.

Entanglement of distillationProposition. Suppose ρA is the reduced density operator of ρAB. For such a ρA one can find a puresuper state |φAB〉 ∈ HA ⊗HB, where ρA is again the corresponding reduced density operator.

This process is called purification. It makes clear that a reduced density operator is not unique. Thestatement follows from the Schmidt decomposition theorem in section 3.1.

Definition 4.3. The entanglement of distillation ED(ρAB) tells us about the number mout of maximalentangled states |φAB〉 that can be purified from nin copies of a given state ρAB (written as (ρAB)⊗nin).More precisely it is defined as the ratio mout over nin in the limit of infinitely many states (ρAB)⊗nin .

ED(ρAB) := limnin→∞

mout

nin(4.14)

A mixed state, that can be distilled, i.e. ED > 0 is called free entangled. Its counterpart is boundentanglement, for ED = 0.

Entanglement costIn contrast to the entanglement of distillation the entanglement cost quantifies the amount of pure-stateentanglement needed to create ρAB using local operations.

Definition 4.4. The entanglement cost EC(ρAB) is therefore defined as

EC(ρAB) := limnout→∞

min

nout(4.15)

where min is the number of maximal entangle states |φAB〉 required to prepare (ρAB)⊗nout .9Mathematically this expression is of course only valid in the sense of definition (4.13).

13

4.3 Generalization of the Schmidt rank 4 MIXED BIPARTITE QUDIT STATES

The entanglement cost has not been computed for any mixed state yet, however the calculation canbe done for our examples, the Bell states.[17]

Example. Let ρp a mixture of the two Bell states |Φ+〉 and |Φ−〉 (see eq. (2.11)), i.e.

ρp ≡ (1− p) |Φ+〉〈Φ+|+ p |Φ−〉〈Φ−| p ∈ [0;1

2]

Then the entanglement cost EC is given by

EC(ρp) = H2(1

2+√p (1− p))

where H2(x) = S∗(x, 1− x) with Shannon entropy S∗(p1, p2) = −∑

i pi log pi.10

Remarkable is that ED and EC are extremal measures, any ’sensible’ entanglement measure shouldlie in between.

Hilbert-Schmidt measureSimilar to the relative entropy of entanglement we can pursue an even more geometrical approach byusing the Hilbert-Schmidt distance ||ρ1 − ρ2||.

Definition 4.5. For an arbitrary state ρAB the Hilbert-Schmidt measure is defined as

EH(ρAB) := minσAB∈S

||ρAB − σAB|| (4.16)

where S is the set of all separable states in HA ⊗HB.

It is intuitively clear that EH serves as a measure of entanglement since it evaluates the distancesto all separable states and takes their minimum.As will be shown in chapter 4.4.4 the Hilbert-Schmidt measure plays an important role connecting

entanglement measures with entanglement witnesses.

4.3 Generalization of the Schmidt rank

Similarly to the generalization approach for the entanglement of formation (4.2.2), one can define theSchmidt number as generalization of the Schmidt rank (discussed in section 3.1).[5]

Definition 4.6. Let ρAB =∑

i piρi, ρi ≡ |ϕkii 〉〈ϕ

kii | be a mixed bipartite state. Every pure state ρi

has its Schmidt rank ki. Set kmax ≡ max ki. The Schmidt number r is defined as the minimum of thehighest Schmidt rank over all possible decompositions.

r := min{dec}

kmax (4.17)

The Schmidt number r cannot be higher than the smaller of the dimensions of the two subsystems.It is called a measure of entanglement and useful in connection with Schmidt witnesses which are

discussed below.

4.4 Entanglement witnesses

A quite geometrical approach is the use of so called entanglement witnesses. Entanglement witnessescover a huge area by themselves. Again this chapter is not meant to be a summary of the whole topicbut specific topics are chosen.A central role plays of course the Entanglement Witness Theorem (EWT).[11]

10Proofing this would need a couple of preparations and is omitted.

14

4.4 Entanglement witnesses 4 MIXED BIPARTITE QUDIT STATES

4.4.1 Entanglement Witness Theorem (EWT)

The Entanglement Witness Theorem states the following:

Theorem 4.2. A state ρent ∈ H is entangled if and only if there exists a Hermitian operator A ∈ Hwhich satisfies

〈ρent, A〉 < 0〈σ,A〉 ≥ 0 ∀σ ∈ S (4.18)

where S is the set of all separable states. A is then called an entanglement witness.

This can be derived via the Hahn-Banach Theorem[12] of functional analysis. A geometrical repre-sentation of this theorem states the following:[10]

Theorem 4.3. Let C be a convex closed set in a topological vector space X, and let b ∈ X but b /∈ C.Then there exists a hyperplane that separates b from the set C.

Proof. of Theorem 4.2. If we identify the above vector spacesX ≡ H and sets C ≡ S we know accordingTheorem 4.3 that there exists a hyperplane which separates entangled and separable states. Since sucha hyperplane must exist there exists an A ∈ H which satisfies 〈ρ,A〉 = 0 where ρ is an arbitrary vectorin H. Theorem 4.2 follows straightaway with figure 4.1a (exchange A by −A if necessary).

Definition 4.7. An entanglement witness A is called optimal, that is Aopt, if there exists a separablestate σ ∈ S such that

〈σ, Aopt〉 = 0 (4.19)

It is called optimal since it detects more entangled states than normal witnesses would do as figure4.1b illustrates.

(a) A plane in Euclidean space (b) An optimal entanglement witness.

Figure 4.1: Geometrical illustration of entanglement witnesses, Source: [10]

4.4.2 Positive Map Theorem (PMT)

From the EWT another useful theorem can be derived, namely the Positive Map Theorem (PMT):

Theorem 4.4. Let Λ be a positive map acting on the Hilbert space. Then a bipartite state ρ is separableif and only if

(I⊗ Λ)ρ ≥ 0 ∀ positivemapsΛ. (4.20)

15


Proof. The ⇐=- direction is shown easily.[10] One applies (I⊗ Λ) to a separable state 2.7 and gets

(I⊗ Λ)ρ =∑i

piρAi ⊗ ΛρBi .

Since Λ is positive, ΛρBi is positive as well and therefore (I⊗ Λ)ρ is positive.The proof for the =⇒ - direction is more complex and provided by Michael, Pawel and Ryszard

Horodecki [8].

In other words that means if we can find a positive map Λ where (I⊗Λ)ρ < 0, i.e Λ is not completelypositive11, we know that ρ is entangled.

Example. An often stated example is the partial transposition PT : (I ⊗ T ). Let’s test it on theBell state |Φ+〉 (see eq. (2.11)). The matrix representation of its corresponding density operatorρB ≡ |Φ+〉〈Φ+| and its partial transposition reads

(I⊗ T )ρB = (I⊗ T )

12 0 0 1

20 0 0 00 0 0 012 0 0 1

2

=

12 0 0 00 0 1

2 00 1

2 0 00 0 0 1

2

.

Calculating the eigenvalues leads to {−12 ,

12 ,

12 ,

12} which means (I⊗ T )ρB is not positive, therefore ρB

is entangled (as we already know).

4.4.3 Positive Partial Transpose (PPT) Criterion

Theorem 4.4 can be used to create a more specific criterion for the special case of a bipartite qubitsystem. It is a necessary and sufficient criterion developed by Peres and Horodecki [8].

Theorem 4.5. A state ρ acting on H2⊗H2, H2⊗H3 or H3⊗H2 is separable if and only if its partialtransposition ρTB is a positive operator, i.e.

ρTB ≡ (I⊗ T )ρ ≥ 0. (4.21)

Definition 4.8. A state satisfying eq. (4.21) is called a PPT state.

The fact that this is indeed a necessary condition can immediately be seen recalling the PMT(Theorem 4.4) and the stated example of the partial transposition. As shown in [10], to prove that thecriterion is also sufficient, one needs a theorem by Stormer and Woronowitz [14],[19] which is repeatedhere for the sake of completeness.

Theorem 4.6. Any positive map Λ that maps operators on Hilbert spaces H2⊗H2, H2⊗H3 or H3⊗H2

can be decomposed in the following way

Λ = ΛCP1 + ΛCP2 T (4.22)

where ΛCP1 and ΛCP2 are completely positive maps.

Proof. Let ρ be a state which satisfies (I ⊗ T )ρ ≥ 0. Since ΛCP1 and ΛCP2 are completely positive itmust be true that

(I⊗ ΛCP1 )ρ+ (I⊗ ΛCP2 )(I⊗ T )ρ ≥ 0

11A positive map Λ is called completely positive if the map (Id ⊗ Λ) is still a positive map ∀d.

16


which can obviously be written as

(I⊗ (ΛCP1 + ΛCP2 T ))ρ ≥ 0 (4.23)

. Now the need of Theorem 4.6 gets apparent. It tells us that eq. (4.23) reduces to

(I⊗ Λ)ρ ≥ 0

where Λ is a positive map. Now since any Λ can be decomposed in terms of (4.22) the PMT (theorem4.4) can be used which tells us that ρ is separable if and only if (I⊗ T )ρ ≥ 0.

4.4.4 Bertlmann-Narnhofer-Thirring Theorem

The Bertlmann-Narnhofer-Thirring Theorem connects the Hilbert-Schmidt measure defined in section4.2.2 with the concept of entanglement witnesses. More specific it tells that the Hilbert-Schmidtmeasure of an entangled state equals the maximal violation of an inequality which basically definesthe entanglement witnesses.Before going into details we will proof a Lemma which will considerably ease the proof of the theorem

itself. After that the mentioned inequality will be defined and the theorem will be stated.

Lemma. Let ρ1 and ρ2 be two arbitrary states and an operator C be defined as12

C(ρ1, ρ2) :=ρ1 − ρ2 − 〈ρ1, ρ1 − ρ2〉 I

||ρ1 − ρ2||(4.24)

Also let ρ0 be the ’nearest’ separable state of ρent, i.e.

||ρ0 − ρent|| = minσ∈S||σ − ρent||

where S denotes again the set of all separable states, then C := C(ρ0, ρent) is an optimal entanglementwitness of ρent as defined by eq. (4.18) and (4.19).

Proof. In order to be an entanglement witness of ρent the operator C has to fulfill 〈ρent, C〉 < 0 and〈σ, C〉 ≥ 0∀σ ∈ S. That this is satisfied can be shown in the following manner:Looking at figure 4.2 it is obvious that

〈ρp − ρ1,ρ1 − ρ2

||ρ1 − ρ2||〉 = 0

defines a hyperplane through ρ1 orthogonal to ρ1 − ρ2.By evaluating 〈ρ, C(ρ1, ρ2)〉, which is

〈ρ, C(ρ1, ρ2)〉 = 〈ρ, ρ1 − ρ2

||ρ1 − ρ2||〉 − 〈ρ1,

ρ1 − ρ2

||ρ1 − ρ2||〉〈ρ, I〉︸︷︷︸

=1

= 〈ρ− ρ1,ρ1 − ρ2

||ρ1 − ρ2||〉

one recognizes that the above hyperplane definition reduces to13

〈ρp, C〉 = 0 (4.25)

, similar one defines the left-hand and right-hand states ρl and ρr (see figure 4.2) as:

〈ρl, C〉 < 0

〈ρr, C〉 > 0

12〈., .〉 is defined by eq. (2.10) .13In this context omitting the arguments of C(., .) means evaluating C(ρ1, ρ2).

17


Figure 4.2: A hyperplane orthogonal to ρ1 − ρ2, Source: [10]

Now consider C ≡ C(ρ0, ρent). Since ρ0 is the nearest separable state of ρent it lies on the boundaryof S. Therefore the plane defined by 〈ρp, C〉 = 0 is tangent to S. Also the plane is orthogonal toρent − ρ0. So we have 〈ρent, C〉 < 0, 〈σ, C〉 ≥ 0 ∀σ ∈ S and 〈ρp, C〉 = 0. Thus C is indeed an optimalentanglement witness.

It is not hard to see that the definition (4.18) of an entanglement witness can be rewritten as

〈σ,A〉 − 〈ρent, A〉 ≥ 0 ∀σ ∈ S (4.26)

This inequality is also called a generalized Bell inequality (GBI). One can define the maximal violationof the GBI as follows.

Definition 4.9. The maximal violation of the GBI is given by

B(ρent) := maxA, ||A−aI||≤1

(minσ∈S

(〈σ,A〉 − 〈ρent, A〉))

(4.27)

where a is the coefficient corresponding to the identity matrix when expanding A into the Pauli-Matrixbasis {σi ⊗ σj}. However, the specific choice of the used norm will not be of any importance to us.

Theorem 4.7. The Hilbert-Schmidt measure EH(ρent) of an entangled state ρent equals the maximalviolation of the GBI B(ρent):

EH(ρent) = B(ρent) (4.28)

Proof. The distance ||ρ1 − ρ2|| of two states can be written as

||ρ1 − ρ2|| = 〈ρ1 − ρ2,ρ1 − ρ2

||ρ1 − ρ2||〉 = 〈ρ1 − ρ2,

ρ1 − ρ2

||ρ1 − ρ2||+ c I〉

where c is a complex constant because (ρ1 − ρ2, c I) = c (tr(ρ1)− tr(ρ2)) = 0 since tr(ρ) = 1∀ρ.Choosing c = − 〈ρ1,ρ2−ρ2〉||ρ1−ρ2|| yields per definition to

||ρ1 − ρ2|| = 〈ρ1 − ρ2,ρ1 − ρ2 − 〈ρ1, ρ1 − ρ2〉 I

||ρ1 − ρ2||〉

= 〈ρ1, C(ρ1, ρ2)〉 − 〈ρ2, C(ρ1, ρ2)〉 (4.29)

Using eq. (4.29) the Hilbert-Schmidt measure EH(ρent) can be written as:

EH(ρent) = minσ∈S||ρent − σ|| = ||ρent − ρ0|| = 〈ρ0, C〉 − 〈ρent, C〉 = ♣

18


From the definition of an optimal entanglement witness it is clear that maxA〈ρent, A〉 = 〈ρent, Aopt〉(Aopt has to be parallel to ρent) and 〈ρ0, Aopt〉 = 0 since ρ0 is on the plane. Since C is an optimalentanglement witness we can rewrite EH(ρent) further as

♣ = maxA

(〈ρ0, A)− 〈ρent, A〉) = maxA

(minσ∈S

(〈σ,A)− 〈ρent, A〉))

= B(ρent)

which completes the proof.

19

5 MULTIPARTITE STATES

5 Multipartite states

The classification and quantification of entanglement is in its early stage when it comes to multipartitesystems. Many measures are designed to function in specific domains where they perform successfuloperations. At the same time the same measures might fail badly in other areas.The mathematical treatment of such measures is a delicate matter. The next few sections provide

the definitions of more or less often stated entanglement measures. [1]

5.1 Some entanglement measures

5.1.1 Geometric measure of entanglement

An attempt to quantify the entanglement of a pure state is given by the minimal distance of the statefrom the set of all pure product states Φ.

Definition 5.1. A geometric measure Eg(Ψ) for a pure multipartite state is defined as follows:

Eg(Ψ):=− log maxΦ|〈Ψ|Φ〉|2 (5.1)

As shown in [18] this measure can be extended to an entanglement monotone measure for mixedstates. It is zero for separable states and rises up to its maximum for e.g. the maximal entangledN -particle GHZ state.

5.1.2 Measure of entanglement by Barnum

Another measure was provided by Barnum.[3]

Definition 5.2. A generalized entanglement measure for qubit systems is given by

Egl := 2− 2

N

N∑j=1

tr(ρ2j ) (5.2)

where N is the number of qubits. ρj is the reduced density matrix of the j-th qubit.Remembering the consequence of the Schmidt decomposition for pure bipartite states (section 3.1),

which stated that tr((ρA)2) = 1 for non entangled states, eq. 5.2 looks like an average of suchevaluations. However, a deeper connection would remain to be shown.

5.2 An entanglement witness

The concept of entanglement witnesses can in principle be extended to general multipartite systemssince the proof done in section 4.4.1 is independent of the dimension of the system. Of course, inpractice it is quite hard to find useful quantities.In [9] the hermitian operator W in an N -partite system is studied:

W =1

2

∑~k∈{0,1}N

b~k (Q+~k−Q−~k ) (5.3)

where Q±~k are projectors to generalized GHZ states, i.e. Q±~k = |G±~k 〉〈G±~k| with

G±~k=

1√2

(|~k〉 ± σ⊗Nx |~k〉

)(5.4)

and b~k are constrained coefficients.

20

5.2 An entanglement witness 5 MULTIPARTITE STATES

It turns out that under the constraint (5.5) the hermitian operator W witnesses entanglement.∑~k

|b~k| ≤ 2N (5.5)

By choosing different sets of b~k one can obtain specific W ’s for different purposes.

21

6 THE CHOICE OF FACTORIZATION

6 The choice of factorization

Until now when we spoke e.g. about a bipartite qubit system in a 4-dimensional Hilbert space H2⊗H2

we silently ignored the specific structure of these tensor factors. Mostly it was implied that the localmeasurement is a projective measurement to the |0〉 state, which has of course its meaning, becausethat is how it’s mainly done in experiments. However, that is certainly a special case. In general it isallowed to do measurements w.r.t. all states in the eigenspace of a particular Hamilton operator thatrelates to the actual present physical problem.We will lift this restriction now and study what would happen to states (respectively entanglement)

if we chose different factorization of the tensor algebra. Among other things we will find that for anentangled state we can always find a factorization of its algebra in which this same state does notappear to be entangled but not the other way around!

6.1 The choice of factorization for pure states

For pure states the situation is quite clear as can be seen in the following section.[15]

Theorem 6.1. For any pure state ρ ∈ H with H being a d1 × d2-dimensional Hilbert-Schmidt spaceone can find a factorization of H, H = A1 ⊗ A1, such that ρ is separable w.r.t. this factorization aswell as a factorization H = B1 ⊗ B2 such that ρ is maximal entangled.

Proof. To see this it suffices to know that for any |Ψ〉 ∈ H there exists a unitary operator U whichtransforms |Ψ〉 in the following way:

U |Ψ〉 =1√d

∑i

|Ψ1,i〉 ⊗ |Ψ2,i〉 (6.1)

with d = min(d1, d2). Let’s consider |Ψ〉 to be separable w.r.t. H = A1 ⊗ A1. So with unitarytransformations we can transform every separable state in a maximal entangled one. Since ρsep =|Ψ〉〈Ψ| the transformed state has the form ρent = UρsepU

† ∈ A1⊗A1. Instead of the state ρ we can aswell transform the underlying algebra: B ⊗ B2 = U(A1 ⊗ A1)U †. Now it becomes clear that ρ whichappears to be separable w.r.t H = A1 ⊗A1 is maximal entangled w.r.t H = B1 ⊗ B2.

6.2 The choice of factorization for mixed states

For mixed states the situation is not that easy and in fact much more interesting. To proof theorem6.2 we need a maximal entangled ONB for a d× d-dimensional Hilbert space. The state (6.2) providesone which is proven in lemma 6.1.[15]

Lemma 6.1. The set of vectors

|χkl〉 =∑j

e2πidjl|ϕj〉 ⊗ |ψj+k〉, j = 1, ..., d; k = 1, ..., d (6.2)

is a maximal entangled ONB of the d× d-dimensional Hilbert space Hd ⊗Hd.

Proof. Evaluating 〈χkl|χmn〉 yields to

〈χkl|χmn〉 =∑i,j

e−2πidjle

2πidin (〈ϕj | ⊗ 〈ψj+k|) (〈ϕi| ⊗ 〈ψi+m|)

=∑i,j

e2πid

(lj−ni) 〈ϕj |ϕi〉︸︷︷︸=δij

〈ψj+k|ψi+m〉︸︷︷︸=δ(j+k),(i+m)

= δkm∑j

e2πidj(l−n) = δkmδln

22

6.2 The choice of factorization for mixed states 6 THE CHOICE OF FACTORIZATION

which proves that the set of |χkl〉’s is an ONB.That these states are entangled can be seen by considering the Schmidt decomposition (see section

3.1). The Schmidt rank equals its maximum value.

Theorem 6.2. For any mixed state ρ ∈ H with H being a d1 × d2-dimensional Hilbert-Schmidt spaceone can find a factorization of H, H = A1 ⊗A1, such that ρ is separable w.r.t. this factorization. Afactorization H = B1 ⊗ B2 such that ρ is entangled exists only beyond a certain bound of mixedness.

Proof. To realize the first statement one has to recall that ρ can be written as∑

α pα|χα〉〈χα| where|χα〉 can be an arbitrary state. We’ve already mentioned that every pure state can be transformed intoa separable state with a unitary transformation U : U |χα〉 = |ϕi,α〉 ⊗ |ψj,α〉. Transforming ρ yields to

UρU † =∑α

pα|ϕi,α〉〈ϕi,α| ⊗ |ψj,α〉〈ψj,α| (6.3)

which is separable.On the other hand we can choose a unitary transformation U which transforms every state in a

maximal entangled one: U |χα〉 = |χkl〉 where |χkl〉 =∑

j e2πidjl|ϕj〉 ⊗ |ψj+k〉 (see Lemma 6.1). The

transformed state ρ is a so called Weyl state:

UρU † =∑k,j

pk|ϕj〉〈ϕj | ⊗ |ψj+k〉〈ψj+k| (6.4)

This is the expansion of UρU † into a set of maximal entangled states. But this set is not convex,therefore the state has not necessarily to be entangled.

Knowing this it makes sense to introduce the following definition:

Definition 6.1. If a state ρ remains separable for any factorization of its algebra (that is for anyunitary transformation U of ρ) it is called a absolutely separable state.

In [15] this theorem is stated more precisely for e.g. Werner states ρw in d× d dimensions:

ρw = αP +1− αd2

Id2 0≤ α ≤1 (6.5)

where P is a projector to a maximal entangled state. This state is entangled for α > 1d+1 . It is shown

that if the largest eigenvalue λ1 of ρw satisfies λ1 >3d2

then there exists always an algebra such thatρw appears entangled.Furthermore the following theorem has been proven for general mixed states in 2× 2 dimensions:

Theorem 6.3. Let ρ be an arbitrary mixed state in 2×2 dimensions with an ordered spectrum {λi}, i =1, .., 4 with λ1 ≥ λ2 ≥ λ3 ≥ λ4. If the eigenvalues satisfy

λ1 − λ3 − 2√λ2λ4 ≤ 0 (6.6)

then ρ is absolutely separable.

23

7 CONCLUSION

7 Conclusion

Let’s summarize the previous steps.We started studying the simplest non trivial case, a pure bipartite state.For these states concrete methods are available to exactly identify entanglement. Looking at the

Schmidt decomposition |ΨAB〉 =∑k

n=1

√pn|uAn , wBn 〉 we derived that one just has to evaluate the trace

of the squared reduced density operator to detect entanglement. Also the Schmidt rank quantifiesentanglement in a certain sense.It was shown that the Von-Neumann entropy S(ρ) = −tr(ρ log ρ) as generalization of the entropy S

from thermodynamics (and therefore leading to a measure of information) offers useful properties byconsidering sub systems and was our first candidate for an entanglement measure.In mixed bipartite states we generalized the idea of the entropy and defined a general entanglement

measure to not only detect entanglement but to quantify it. We have seen that there is no uniquemeasure and stated a few possibilities, each for different purposes.Further we saw that pure systems cannot be entangled with other systems. In this sense the whole

is not the sum of its parts but more:

»Bestmögliches Wissen um ein Ganzes schließt nicht notwendig das Gleiche für seineTeile ein.« - Erwin Schrödinger[13]»The best possible knowledge of a whole does not necessarily include the best possible

knowledge of all its parts.«

From the Hahn-Banach Theorem of functional analysis it follows that there must exist an operatorwhich detects entanglement, namely an entanglement witness, which was discussed as well. An opera-tional criterion, the PPT-criterion, was derived with the help of these witnesses.It was shown that one is even able to construct an optimal entanglement witness that was used

to proof the Bertlmann-Narnhofer-Thirring Theorem which connects the concept of entanglementwitnesses with the approach of entanglement measures.The quantification of multipartite states is even more of a challenge. Mathematically the most

measures are not fully understood in the sense that many properties are just proven for special caseswhich are merely assumed to hold in general. Not only the quantification is a problem: the question,if a state is actually entangled or not, cannot be answered in general. Initial attempts were made toprovide tools, but that hopefully should only be the beginning.Finally the factorization of the algebra was considered. The results were quite interesting. We can

transform every entangled state to a non-entangled one while the reverse is not true in general but justfor certain bound of mixedness.

24

References References

References

[1] L. Amico, R. Fazio, A. Osterloh, and V. Vedral, Entanglement in many-body systems, Reviews ofModern Physics 80 (2008), no. 2, 517.

[2] J. Audretsch, Verschränkte Systeme, Wiley Online Library, 2005.

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