Polygamy of Distributed Entanglement

# Polygamy of Distributed Entanglement

Francesco Buscemi Statistical Laboratory, DPMMS, University of Cambridge, Wilberforce Road, CB3 0WB, UK    Gilad Gour Institute for Quantum Information Science, University of Calgary, Alberta T2N 1N4, Canada Department of Mathematics and Statistics, University of Calgary , Alberta T2N 1N4, Canada    Jeong San Kim Institute for Quantum Information Science, University of Calgary, Alberta T2N 1N4, Canada
August 28, 2019
###### Abstract

While quantum entanglement is known to be monogamous (i.e. shared entanglement is restricted in multi-partite settings), here we show that distributed entanglement (or the potential for entanglement) is by nature polygamous. By establishing the concept of one-way unlocalizable entanglement (UE) and investigating its properties, we provide a polygamy inequality of distributed entanglement in tripartite quantum systems of arbitrary dimension. We also provide a polygamy inequality in multi-qubit systems, and several trade offs between UE and other correlation measures.

###### pacs:
03.67.-a, 03.67.Hk, 03.65.Ud,

## I Introduction

Quantum entanglement is a non-local quantum correlation providing a lot of useful applications in the field of quantum communications and computations such as quantum teleportation and quantum key distribution tele ; qkd1 ; qkd2 . This important role of quantum entanglement has stimulated intensive study in both way of its quantification and qualification.

One of the essential differences of quantum correlations (especially, quantum entanglement) from other classical ones is that it cannot be freely shared among the parties in multipartite quantum systems. In particular, a pair of components that are maximally entangled cannot share entanglement CKW ; ov nor classical correlations KW with any part of the rest of the system, hence the term Monogamy of Entanglement (MoE) T04 . Monogamy of entanglement was shown to have a complete mathematical characterization for multi-qubit systems ov using a certain entanglement measures, the concurrence ww .

Whereas MoE shows the restricted sharability of multi-party quantum entanglement, the distribution of entanglement, or Entanglement of Assistance (EoA) d ; cohen in multipartite quantum systems was shown to have a dually monogamous (or Polygamous) property. Using Concurrence of Assistance (CoA) lve as the measure of distributed entanglement, it was also shown that whereas monogamy of entanglement inequalities provide an upper bound for bipartite sharability of entanglement in a multipartite system, the same quantity provides a lower bound for distribution of bipartite entanglement in a multipartite system gbs . In this paper, by introducing the concept of One-way Unlocalizable Entanglement (UE), we provide a polygamy inequality of entanglement in tripartite quantum systems of arbitrary dimension using entropic entanglement measure. Based on the functional relation between concurrence and entropic measure in two-qubit systems, we provide a polygamy inequality in multi-qubit systems. We also provide several trade offs between UE and other correlations such as EoA, and localizable entanglement.

The paper is organized as follows. In Sec. II, we provide the definition of UE, and its basic properties. In Sec. III, we provide a polygamy inequality of distributed entanglement in tripartite quantum systems in terms of entropy and EoA. In Sec. IV, we generalize the polygamy inequality of entanglement into multi-qubit systems, and provide a more tight polygamy inequality for three-qubit systems. In Sec. V, we provide several trade offs between UE and other correlations, and we summarize our results, in Sec. VI.

## Ii One-Way Unlocalizable Entanglement

### ii.1 Definition

For any bipartite quantum state , its one-way distillable common randomness DV is defined as

 C←D(ρAB)=limn→∞1nI←(ρ⊗nAB), (1)

where, the function  Henderson-Vedral01 is

 I←(ρAB)=max{Mx}[S(ρA)−∑xpxS(ρxA)], (2)

and where the maximum is taken over all the measurements applied on system . Here, is the von Neumann entropy of , is the probability of the outcome , and is the state of system when the outcome was .

For a tripartite pure state with , , and , it was shown that KW

 S(ρA) =I←(ρAB)+Ef(ρAC). (3)

Here, is the Entanglement of Formation (EoF) of defined as bdsw

 Ef(ρAC)=min∑ipiS(ρiA), (4)

where the minimization is taken over all pure state decomposition of such that,

 ρAC=∑ipi|ϕi⟩AC⟨ϕi|, (5)

with .

As a dual quantity to EoF, EoA is defined by the maximum average entanglement of ,

 Ea(ρAC)=max∑ipiS(ρiA), (6)

over all possible pure state decompositions of .

###### Definition 1.

The one-way unlocalizable entanglement (UE) of a bipartite state is defined as follows:

 E←u(ρAB):=S(ρA)−Ea(ρAC), (7)

where denotes the reduced state of a purification of .

The one-way unlocalizable entanglement can be equivalently characterized as follows:

###### Lemma 1.

For any given bipartite state , its one-way unlocalizable entanglement is given by

 E←u(ρAB)=min{Mx}[S(ρA)−∑xpxS(ρxA)], (8)

where the minimum is taken over all possible rank-1 measurements applied on subsystem .

###### Proof.

Eq. (8) can be rewritten as

 E←u(ρAB)=S(ρA)−max{Mx}∑xpxS(ρxA), (9)

where the maximum is taken over all possible rank-1 measurements applied on system .

Since is a pure state, all possible pure state decompositions of can be realized by rank-1 measurements of subsystem , and conversely, any rank-1 measurement can be induced from a pure state decomposition of . Thus, the second term on the right hand side of Eq. (9) is the maximum average entanglement over all possible pure state decomposition of , which is the definition of , and this completes the proof. ∎

By definition, the UE of is the difference between and . Here, quantifies the entanglement of the pure state with respect to the bipartite cut , whereas measures the maximum average entanglement that can be localized on the subsystem with the assistance of . The terminology used is then clear. Figure 1 graphically illustrates this separation.

### ii.2 Properties

###### Lemma 2.

For all bipartite states and ,

 E←u(ρAB⊗σA′B′)≤E←u(ρAB)+E←u(σA′B′), (10)

where

 E←u(ρAB⊗σA′B′)=min{Lz}[S(ρA⊗σA′)−∑zrzS(τzAA′)], (11)

with , , and the minimum is taken over all possible rank-1 measurements applied on subsystem .

###### Proof.

Let and be the optimal rank-1 measurements on subsystems and for and respectively, then, we have

 E←u (ρAB)+E←u(σA′B′) =S(ρA)+S(σA′)−∑xpxS(ρxA)−∑yqyS(σyA′) =S(ρA⊗σA′)−∑xypxqyS(ρxA⊗σyA′) ≥E←u(ρAB⊗σA′B′), (12)

where , , and the second equality is due to the additivity of von Neumann entropy and the definition of . ∎

By Lemma 2, we can assure the existence of the regularized UE

 E←u,∞(ρAB):=limn→∞E←u(ρ⊗nAB)n, (13)

which satisfies

 E←u,∞(ρAB)≤E←u(ρAB). (14)

#### ii.2.2 Simple Lower Bound

###### Lemma 3.

For any bipartite state ,

 E←u(ρAB)≥max{I←c(ρAB), 0}, (15)

where is the coherent information of .

###### Proof.

Let be a purification of , then due to the monotonicity of entanglement, we have

 Ea(ρAC)≤min{S(ρA), S(ρC)}, (16)

where .

Thus, together with Lemma 1, we have

 E←u(ρAB) =S(ρA)−Ea(ρAC) ≥max{S(ρA)−S(ρC), 0} =max{I←c(ρAB), 0}, (17)

where the last equality is due to the purity of , that is, . ∎

Since is a purification of both and , we have

 E←u(ρ⊗nAB)+Ea(ρ⊗nAC)=nS(ρA). (18)

By taking the limit , and due to the relation Smolin-Ver-Win

 limn→∞Ea(ρ⊗nAC)n=min{S(ρA),S(ρC)}, (19)

we have that

 E←u,∞(ρAB)=max{I←c(ρAB),0}. (20)

Eq. (20) implies that, in the asymptotic limit of many copies, separable states do not exhibit quantumness in their correlations, or their correlations are completely erasable. This is a strong evidence that the distinction between separable and entangled states is operational only in asymptotic sense, since separable states can exhibit non-zero UE in finite case.

## Iii Polygamy of entanglement in tripartite quantum systems

For any bipartite pure state , its concurrence, is defined as ww

 C(|ϕ⟩AB)=√2(1−trρ2A), (21)

where . For any mixed state , its concurrence is defined via convex-roof extension, that is,

 (22)

where the minimum is taken over all possible pure state decompositions, .

As a dual value to concurrence, CoA lve of is defined as

 (23)

where the maximum is taken over all possible pure state decompositions of .

By using concurrence and CoA as the quantification of bipartite entanglement, it was shown that there exists a polygamy relation of entanglement in multi-qubit systems gbs . More precisely, for any pure state in an -qubit system where for ,

 C2A1(A2⋯An)≤(CaA1A2)2+⋯+(CaA1An)2, (24)

where is the concurrence of with respect to the bipartite cut and , and is the CoA of for .

In this section, we provide an analytic upper bound of UE in Eq. (8), and derive a polygamy inequality of entanglement in terms of von-Neumann entropy and EoA for tripartite quantum systems of arbitrary dimension.

First, for an upper bound of UE, we have the following theorem.

###### Theorem 4.

For any bipartite state in a bipartite quantum system ,

 E←u(ρAB)≤I(ρAB)2, (25)

where is the mutual information of .

###### Proof.

Let be a spectral decomposition of where is the dimension of the subsystem . The proof method follows the construction used in christandl .

For any state , define the channels

 M0(σ): =dB−1∑i=0|ei⟩⟨ei|σ|ei⟩⟨ei| M1(σ): =dB−1∑i=0|~ej⟩⟨~ej|σ|~ej⟩⟨~ej|, (26)

where is the Fourier basis such that,

 |~ej⟩=1√ddB−1∑k=0ωjkd|ek⟩, j=0,…,dB−1, (27)

and is the -th root of unity.

Notice that , and , so that . We can also write

 M0(σ)=1dBdB−1∑b=0ZbσZ−b, M1(σ)=1dBdB−1∑a=0XaσX−a, (28)

where and are generalized -dimensional Pauli operators,

 Z= dB−1∑j=0ωjd∣∣ej⟩⟨ej∣∣, X= dB−1∑j=0∣∣ej+1⟩⟨ej∣∣=dB−1∑j=0ω−jd|~ej⟩⟨~ej|. (29)

In the following, we will write

 (IA⊗M0)(ρAB) =dB−1∑i=0σiA⊗λi|ei⟩B⟨ei|, (IA⊗M1)(ρAB) =dB−1∑j=0τjA⊗1dB|~ej⟩B⟨~ej|, (30)

where , and for .

The induced ensembles on by the channels and will be denoted by and , and the entropy defects of the induced ensembles on will be denoted as

 χ(E0)= S(ρA)−dB−1∑i=0λiS(σiA), χ(E1)= S(ρA)−1dBdB−1∑i=0S(τjA). (31)

By defining a four-partite quantum state in such that

 ΩXYAB:=1d2BdB−1∑x,y=0|x⟩X⟨x|⊗|y⟩Y⟨y|⊗(IA⊗XxBZyB)ρAB(IA⊗Z−yBX−xB), (32)

we have

 ΩXAB= 1dBdB−1∑x=0|x⟩X⟨x|⊗XxB(dB−1∑i=0σiA⊗λi|ei⟩B⟨ei|)X−xB, ΩYAB= 1dBdB−1∑y=0|y⟩Y⟨y|⊗ZyB(dB−1∑j=0τjA⊗1dB|~ej⟩B⟨~ej|)Z−yB, (33)

and

 ΩAB=ρA⊗IBdB. (34)

By straightforward calculation, we can obtain

 I(ΩX(AB))= S(ΩX)+S(ΩAB)−S(ΩXAB) = logdB+logdB+S(ρA)−logdB −S(∑iσiA⊗λi|ei⟩B⟨ei|) = logdB+S(ρA)−H(→λ)−∑iλiS(σiA) = logdB−S(ρB)+χ(E0), (35)

where is the mutual information of with respect to the bipartite cut , and the second, third equalities are due to the joint entropy theorem nc . Analogously, we have

 I(ΩY(AB))=χ(E1), I(Ω(XY)(AB))=logdB+S(ρA)−S(ρAB) =logdB+I←c(ρAB). (36)

Due to the independence of subsystems and , we have , which implies

 χ(E0)+χ(E1)≤I(ρAB). (37)

Since and of Eq. (37) can be obtained, respectively, from by rank-1 measurements and of subsystem , by defining a rank-1 measurement

 {|ei⟩B⟨ei|2,|~ej⟩B⟨~ej|2}i,j, (38)

we have

 E←u(ρAB)≤χ(E0)2+χ(E1)2≤I(ρAB)2, (39)

which completes the proof. ∎

###### Corollary 1.

For any tripartite pure state , we have

 S(ρA)≤Ea(ρAB)+Ea(ρAC). (40)
###### Proof.

By Lemma 7, we have

 Ea(ρAC) =S(ρA)−E←u(ρAB), Ea(ρAB) =S(ρA)−E←u(ρAC), (41)

and thus,

 Ea(ρAC)+Ea(ρAB)=2S(ρA)−E←u(ρAB)−E←u(ρAC). (42)

Now, by Theorem 4, we have

 Ea(ρAC)+ Ea(ρAB) ≥ 2S(ρA)−I(ρAB)2−I(ρAC)2 = 2S(ρA)−S(ρA)/2−S(ρB)/2+S(ρAB)/2 −S(ρA)/2−S(ρC)/2+S(ρAC)/2 = S(ρA). (43)

Corollary 1 tells us that for a tripartite pure state of arbitrary dimension, there exists a polygamy relation of entanglement in terms of entropy of entanglement and EoA. Furthermore, this is, we believe, the first result of the polygamous (or dually monogamous) property of distribution of entanglement in multipartite higher-dimensional quantum systems rather than qubits.

## Iv Polygamy relation of entanglement in multi-qubit quantum systems

In this section, we show that the polygamy inequality of entanglement in Corollary 1 can be generalized into multipartite quantum systems for the case when each subsystem is a two-level quantum system. By investigating the functional relation between concurrence and EoF in two-qubit systems ww , we show that there exists a polygamy inequality of entanglement in terms of entropy and EoA in -qubit systems. We also show that, in three-qubit systems, we have a more tight polygamy inequality than Eq. (40) in Corollary 1.

First, let us consider the functional relation of concurrence with EoF in two-qubit systems. For a 2-qubit mixed state (or a pure state ), the relation between its concurrence, and can be given as a monotone increasing, convex function  ww , such that

 Ef(ρAB)=E(CAB), (44)

where

 (45)

and is the binary entropy function . The same function relates also the EoA of a bipartite state with its CoA via the equation

 Ea(ρAB)≥E(CaAB), (46)

which is due to the convexity of and the definition of EoA. The following lemma shows an important property of the function .

###### Lemma 5.
 E(√x2+y2)≤E(x)+E(y), (47)

for such that .

###### Proof.

By considering

 f(x,y)=E(x)+E(y)−E(√x2+y2), (48)

as a two-vairable real-valued function on the domain , it is enough to show that in .

Since is a compact subset in , whereas is analytic on the interior of , and continuous on , the minimum value of arises only on the critical points or on the boundary of . It can be directly checked that does not have any vanishing gradient on the interior of , and has non-negative function values on the boundary of . Thus, is non-negative on the domain . ∎

### iv.1 Three-qubit systems

A direct observation from CKW shows that, for a 3-qubit pure state ,

 C2A(BC)=C2AB+(CaAC)2, (49)

where and are the concurrence and concurrence of assistance of and respectively. (Later, Eq. (49) was formally shown in ys .) From Eq. (49) together with Lemma 5, we have the following theorem.

###### Theorem 6.

For a three-qubit pure state ,

 S(ρA)≤Ef(ρAB)+Ea(ρAC). (50)
###### Proof.

Since is a bipartite pure state in with respect to the bipartite cut and , we have,

 S(ρA)=Ef(ψA(BC))=E(CA(BC)). (51)

Thus,

 S(ρA) = E(CA(BC)) (52) = E(√C2AB+CaAC2 ) ≤ E(CAB)+E(CaAC) ≤ Ef(ρAB)+Ea(ρAC),

where the first inequality is by Lemma 5, and the second inequality is by Eq. (46). ∎

Thus, the polygamy relation of distributed entanglement in tripartite quantum systems obtained in Corollary 1 can have a more tight form in three-qubit systems. Furthermore, the result of Theorem 6 together with Eqs. (3) and (7) give us the following corollary.

###### Corollary 2.

For any two-qubit mixed state with rank less than or equal to two,

 I←(ρAB)≤Ea(ρAB), (53)
 E←u(ρAB)≤Ef(ρAB). (54)
###### Remark 1.

Eq. (54) of Corollary 2 implies that any two-qubit separable state of rank less than or equal to two has zero UE, . However, this is not generally true for two-qubit separable states of rank larger than two. Here, we provide an example of two-qubit rank-three separable state with non-zero UE.

Example: Let us consider the following state in quantum system CCJKKL ,

 (55)

where and are two orthogonal states in the such that

 |x⟩= (|02⟩+√2|10⟩)/√3, |y⟩= (|12⟩+√2|01⟩)/√3. (56)

First, since , it is clear that , therefore we have .

Since , Hughston-Jozsa-Wootters (HJW) theorem HJW says that for any decompositions of , there exists an unitary operator such that with . Thus,

 16pi(|ui1|2+2|ui2|2ui1u∗i2ui2u∗i1|ui2|2+2|ui1|2) = 13IA+13|ψi⟩A⟨ψi|, (57)

with , and we obtain that for any pure state in any pure state decomposition of .

Since is a pure state, we have

 Ef(|ϕi⟩AC) = E(C(|ϕi⟩AC)) (58) = H(23) = log23−23,

and thus .

Now, we have , whereas, it can be easily seen that has a Positive Partial Transposition (PPT) which is equivalent to separability for two-qubit states horo1 . Thus, is a two-qubit, rank-three separable state with non-zero UE.

### iv.2 n-qubit systems

The polygamy inequality of entanglement in -qubit systems in Eq. (24) gives us an inequality

 CA1(A2⋯An)≤√(CaA1A2)2+⋯+(CaA1An)2. (59)

Thus, together with Lemma 5, we have the following theorem.

###### Theorem 7.

For any -qubit pure state ,

 S(ρA1)≤Ea(ρA1A2)+⋯+Ea(ρA1An). (60)
###### Proof.

First, let us assume that , then we have

 S(ρA1)= E(CA1(A2⋯An)) ≤ E(√(CaA1A2)2+⋯+(CaA1An)2) ≤ E(CaA1A2)+E(√(CaA1A3)2+⋯+(CaA1An)2) ≤ E(CaA1A2)+E(CaA1A3)+⋯+E(CaA1An) ≤ Ea(ρA1A2)+⋯+Ea(ρA1An), (61)

where the first inequality is due to the monotonicity of the function , the second and third inequalities are obtained by iterating Lemma 5, and the last inequality is by Eq. (46).

Now, assume that