teresult

Physics Letters A 330 (2004) 418–423

www.elsevier.com/locate/pla

Comparability of multipartite entanglement✩

Zhengfeng Ji∗, Runyao Duan, Mingsheng Ying

State Key Laboratory of Intelligent Technology and Systems,Department of Computer Science and Technology Tsinghua University, Beijing 100084, China

Received 30 June 2004; received in revised form 26 July 2004; accepted 13 August 2004

Available online 26 August 2004

Communicated by P.R. Holland

Abstract

We prove, in a multipartite setting, that it is always feasible to exactly transform a genuinelym-partite entangled pure stawith sufficient many copies to any otherm-partite state via local quantum operation and classical communication. Thisaffirms the comparability of multipartite entangled pure states. 2004 Elsevier B.V. All rights reserved.

PACS:03.67.Hk; 03.67.-a

Keywords:Local operation and classical communication; Multipartite entanglement; Teleportation; Entanglement exchange rate

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1. Introduction

Due to its numerous important applications[1–5]in the area of quantum information and computatientanglement has been extensively studied in thedecade. The concept of considering entanglement aa new type of useful resource has been widelycepted. Efforts have been made to quantitativelyqualitatively measure this kind of resource in both

✩ This work was partly supported by the National FoundationNatural Sciences of China (Grant No. 60273003).

* Corresponding author.E-mail addresses:jizhengfeng98@mails.tsinghua.edu.cn

(Z. Ji), dry02@mails.tsinghua.edu.cn(R. Duan),yingmsh@mail.tsinghua.edu.cn(M. Ying).

0375-9601/$ – see front matter 2004 Elsevier B.V. All rights reserveddoi:10.1016/j.physleta.2004.08.015

t

partite and multipartite cases[6–8]. In the bipartitecase, a unique measure has been found[6] to quantifyentanglement asymptotically. However, it is a muchmore complicated problem for multipartite entangments even in the asymptotic manner[7,9–12].

Entanglement transformations can often be utilizto define corresponding measures of entanglementassign a same scalar number to two different engled states if they can be converted interchangeablto each other by some predefined physical transmation. Here, the reversibility plays an important roin the quantification procedure which acts as a bance weighing these two states. Moreover, standweights (some standard entangled states, for instathe maximally entangled states or minimal reversientanglement generating set (MREGS)[7]) are often

.

Z. Ji et al. / Physics Letters A 330 (2004) 418–423 419

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used to measure entanglements in such a schemethe structure of entanglement is so complicated thait is difficult to define proper entanglement measubased on this interconvertible scheme except for ssimple cases.

Various approaches can be adopted to copethis difficulty. One common method is to use muticomponent measures. For example, in the bipacase, Nielsen’s criteria of exact LOCC transformat[13] indicates that the vector of Schmidt coefficieninstead of any scalar, is a proper entanglement msure when exact transformations are considered.of the disadvantages of this approach is that thereist incomparable states due to lack of total ordermulticomponent measures[11,12]. Thus, they maylead to complicated, or even infinite, classificatioof entanglement. Another approach that we are ccerned with is to measure entanglement relativelythis approach, the requirement of reversibility is dcarded and there are no so called standard entaments. Instead, we focus on how many target engled states can be obtained asymptotically, taking bothinitial state and target state as parameters. Specificwe define the entanglement exchange rate between|ψ〉and|φ〉 as1

R(|ψ〉, |φ〉)

(1)= sup

{m

n: |ψ〉⊗n is convertible to|φ〉⊗m

}.

One can see that entanglement exchange rafact generalizes entanglement measure by permiquantification using entanglement transformation tare not interconvertible. This modification allowsstudy entanglement quantitatively at a more extendegree. The scheme of defining entanglement ofmation and entanglement of distillation[14] separatelyis also capable of describing the non-interconvertphenomena in the bipartite case. However, ourproach is more flexible and has a different physimeaning. To emphasize the main difference betwentanglement exchange rate and entanglementsure, we first express entanglement measure in

1 This definition is borrowed from the concept of exchange rin finance and we adopt indirect quotation notation to make it cpatible with the basic requirement of entanglement measure: tonote more entanglements with larger values.

t,

-

-

form of entanglement exchange rate. Specifically,entanglement measureE based on interconvertibltransformation induces an entanglement exchangeRE where

(2)RE

(|ψ〉, |φ〉) = E(|ψ〉)E(|φ〉) .

ForRE , the following very strong transitivity

(3)RE

(|ψ0〉, |ψ1〉) × RE

(|ψ1〉, |ψ2〉) = RE

(|ψ0〉, |ψ2〉)

always holds while for general entanglement excharateR we only have a weaker transitivity

(4)R(|ψ0〉, |ψ1〉

) × R(|ψ1〉, |ψ2〉

)� R

(|ψ0〉, |ψ2〉)

which is obvious since for most transformations ofterest, their combination is still the same type of traformation. This inequality can be interpreted that, jas in economics,bid/offer spreadmay also occur inentanglement exchange process.

The perspective of regarding entanglement asful resource leads us to the question of whethertanglements are interchangeable. One might exthat any entangled states, which might be usefusome quantum tasks, can be obtained starting fsome special types of entanglement. This requiremcan be briefly expressed asR(|ψ〉, |φ〉) > 0 which isproved to be true when the initial state|ψ〉 is genuinelym-partite entangled. That is, it is always feasibleexactly transform an entangled pure states with sucient many copies to any other entangled state. Fobipartite case, this assertion is just a simple corolof the Nielsen’s theorem[13]. In order to warrant theapproach of entanglement exchange rate also win multipartite case, it is needed to verify the aboassertion form-partite states withm � 3. The diffi-culty for this purpose is that we have no longerNielsen’s theorem and even the Schmidt decomption for the multipartite case. In this Letter, we are ato prove that for anym � 2, eachm-partite entangledstate with sufficient many copies can be transformto any otherm-partite state. And we organize this pper as follows. First, entanglements that are genuinm-partite entangled (m-partite entangled for short) arformally defined. Then we give the key lemma of thpaper which correlates the multipartite and biparcase. Finally, we prove the main result using thelemma and conclude briefly.

420 Z. Ji et al. / Physics Letters A 330 (2004) 418–423

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2. Comparability of pure entanglements

Entanglements that are genuinely multipartitetangled are states that cannot be written in prodform between any bipartite partition of the partieWith m = 3, for example, the|GHZ〉 [15] and |W 〉state are 3-partite entangled while

(5)|001〉 + |111〉√

2is not if each of the three qubits belongs to a differparty. We can formally describe this concept as flows.

DefineP to be the set of all parties under considation,

(6)P = {Pi, 1� i � m}.A state|ψ〉 shared bym parties inP is calledm-partiteentangled state if for any non-empty proper subof P , sayA, |ψ〉 is entangled according to the bipartpartitionA andP −A.

The elegant majorization criteria[13] indicatesthat, from sufficient many copies of bipartite entangments, any other entanglements are always obtainablvia LOCC only. Though there is no exact countpart of this criteria in the multipartite case, Nielsentheorem is still useful here. It allows us to concetrate bipartite entanglement to Bell states with whteleportation can be done between different partTeleportation back and forth in fact removes the cstrains of LOCC and allows any global operationscost of the consumption of shared entanglements.

Obviously, the main obstacle to generalize tocase ofm � 3 is then how to convert the multipartistates to bipartite entanglement with which the abidea can be realized. The following lemma devotepad this gap.

Lemma 1. Let |ψ〉 be a state shared byn+ m parties,

A1,A2, . . . ,An,B1, . . . ,Bm.

Denote

A = {Ai,1 � i � n},B = {Bj ,1 � j � m}.If |ψ〉 is entangled between the partitionA,B, thereexist somei, j and some LOCC procedure that prduces a pure entangled state betweenAi andBj .

Two claims are given beforehand in the followinto simplify the proof ofLemma 1.

Claim 2. If there exists non-zeroλ such that|φ0〉A|φ1〉B+ λ|ψ0〉A|ψ1〉B is product state between Alice anBob (without normalization), then|〈φ0|ψ0〉| = 1 or|〈φ1|ψ1〉| = 1, that is, |φ0〉, |ψ0〉 or |φ1〉, |ψ1〉 differonly in some global phase.

Proof. Consider two projective measurements incluing

|φ0〉〈φ0|A ⊗ IB

and

|ψ0〉〈ψ0|A ⊗ IB,

as one of the projective operators, respectively. If one(the first, for example) of the projective operators ocurs with zero probability in the corresponding mesurement, then the norm of|φ0〉(|φ1〉+λ〈φ0|ψ0〉|ψ1〉)is zero, which indicates that|φ1〉, |ψ1〉 differ onlyin some global phase. Otherwise, since|φ0〉|φ1〉 +λ|ψ0〉|ψ1〉 is product state, measurements definabove does not change the state of Bob’s side. Tthere exists some non-zero complex numberc suchthat

(7)|φ1〉 + λ〈φ0|ψ0〉|ψ1〉 = c(〈ψ0|φ0〉|φ1〉 + λ|ψ1〉

),

that is,

(8)(1− c〈ψ0|φ0〉

)|φ1〉 = (cλ − λ〈φ0|ψ0〉

)|ψ1〉.|φ1〉, |ψ1〉 are not zero vector, so it is impossible thonly one of 1− c〈ψ0|φ0〉 andcλ − λ〈φ0|ψ0〉 is zero.If both of them are non-zero,|φ1〉, |ψ1〉 differ only insome global phase. Or else both 1−c〈ψ0|φ0〉 andcλ−λ〈φ0|ψ0〉 are zero, then we have

(9)〈ψ0|φ0〉 = 1

c,

(10)〈φ0|ψ0〉 = c.

Thus,

(11)∣∣〈φ0|ψ0〉

∣∣2 = 〈ψ0|φ0〉〈φ0|ψ0〉 = 1.

In this case,|φ0〉, |ψ0〉 differ only in a global phase.�

Z. Ji et al. / Physics Letters A 330 (2004) 418–423 421

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Claim 3. |φ〉AB is bipartite entangled state an|ψ0〉A|ψ1〉B is product state, then there is at most onon-zero realλ such that|φ〉+λ|ψ0〉|ψ1〉 is a productstate.

Proof. If there are two different non-zero realλ, sayλ0, λ1, that satisfy the requirement, we have

(12)|φ〉 + λ0|ψ0〉|ψ1〉 = |α〉|β〉,(13)|φ〉 + λ1|ψ0〉|ψ1〉 = |α′〉|β ′〉.

Minus above two equations we get,

(14)|α′〉|β ′〉 = |α〉|β〉 + (λ1 − λ0)|ψ0〉|ψ1〉.By Claim 2, |α〉, |ψ0〉 or |β〉, |ψ1〉 differ only in aglobal phase which contradicts with the fact that|φ〉is entangled. �

Assisted by above two claims, we are now readyproveLemma 1.

Proof of Lemma 1. We consider projective measurment onA1 and discuss two different cases separatLet |α0〉 be a pure state ofA1 and

P = |α0〉〈α0|.Case1. There exists someP such that(P ⊗ I)|ψ〉

is purely entangled betweenA′ = A − {A1} andB.Extend|α0〉 to a complete orthogonal basis in thed

dimensional state space ofA1, {|αi〉,0 � i < d}, wehave

|ψ〉 =d−1∑i=0

√pi |αi〉|ψi〉,

where |ψ0〉 is entangled betweenA′,B. If there issome |ψi〉 not entangled betweenA′,B, say |ψ1〉,we can change the basis{|αi〉,0 � i < d} properlyto {|α′

i〉,0 � i < d} and make the correspondin|ψ ′

0〉, |ψ ′1〉 both entangled. Specifically, let

|α′0〉 = cosθ |α0〉 + sinθ |α1〉,

(15)|α′1〉 = sinθ |α0〉 − cosθ |α1〉,

(16)|α′i〉 = |αi〉, 2 � i < d.

Then we have[16]

(17)

(√p′

0 |ψ ′0〉√

p′1 |ψ ′

1〉

)=

[cosθ sinθ

sinθ −cosθ

](√p0 |ψ0〉√p1 |ψ1〉

).

EmployingClaim 3, it is easy to see thatθ needed forour purpose exists. Continuously applying the tenique of changing the basis, we can make all the|ψi〉entangled. ThusA1 can do a projective measuremeaccording to the chosen basis, leaving an entanpure state betweenA′,B after the measurement.

Case2. Otherwise, for any state|α0〉, (P ⊗ I)|ψ〉is product pure state betweenA′ andB. Then we canexpand|ψ〉 as

(18)|ψ〉 =d−1∑i=0

√pi |i〉A1|φi〉A′ |ψi〉B.

Moreover, let|α0〉 = |i〉 + λ|j 〉, we get

(19)√

pi |φi〉|ψi〉 + λ√

pj |φj 〉|ψj 〉is also a product state.Claim 2states that for anyi, j ,|φi〉, |φj 〉 or |ψi〉, |ψj 〉 differ only in a global phaseSimple calculation tells that all states in{|φi〉,0 �i < d} or all in {|ψi〉,0 � i < d} differ only in a globalphase. However, the later case is impossible sinccontradicts with the fact that|ψ〉 is entangled betweeA,B. In the former case, the reduced state of|ψ〉 onA1,B1,B2, . . . ,Bm is an entangled pure states.

In both cases above, we managed to reducenumber of parties without destroying the entanglembetweenA,B. Repeat the procedure several times,will be able to find two partiesAi andBj fulfilling ouraim. �

Multiple uses ofLemma 1lead to the main theoremof this Letter:

Theorem 4. For any m-partite entangled pure stat|ψ〉 and anym-partite state|φ〉, it is always feasibleto obtain|φ〉 provided that the available state|ψ〉 aresufficient and exact LOCC are allowed. Or we canbreviate the result using entanglement exchangeasR(|ψ〉, |φ〉) > 0.

Proof. Let P = {Pi,1 � i � m} be them parties andA = {P1}, B = P − A. The proof ofLemma 1indi-cates that there exists somej �= 1 such that|ψ〉 istransformed to a bipartite pure entangled state betwP1 andPj . The value ofj depends on the measurment result of each step. Different possible outcommay lead to differentj and, of course, different typof entanglement betweenP1 and Pj . But regardless

422 Z. Ji et al. / Physics Letters A 330 (2004) 418–423

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of this indeterminism, sufficient manym-partite en-tanglements can produce sufficient many bipartitetanglements between at least one pair of parties,P1 andPj∗ . With these sufficient many bipartite etanglements (that belong to finite possible types)teleportation, all operation on the joint system ofP1andPj∗ can be performed using only LOCC and thuwe can think ofP1 andPj∗ as a single party. The samarguments work until the number of parties decreato one. That is, among them parties, we can performany quantum operations which include the preparaof another state|φ〉. �

3. Conclusions

In sum, we have proved the comparability of geuinely m-partite entangled states in the sense of exLOCC transformation. Thus, as a type of resourgenuinelym-partite entangled states are all qualitively equal, at least theoretically. This affirms tnon-zero property of entanglement exchange ratetroduced in Section1 which might be an indicationthat the approach of entanglement exchange raworth of studying since it simplifies the characterizeentanglement property and the classification of engled states without loss of basic physical backgrounEntanglement exchange rate is itself a rough metstudying entanglement property and cannot descentanglement accurately especially in the multiparcase. However, it helps to classify entanglement prerly using entanglement transformation. At the figlance, this result can be extended to the mixed cwhile in fact such an extension fails to hold becauof the existence of bound entangled states[17]. Thisanalysis tells that pure entangled states differ quantively while mixed entangled states differ qualitativein our interpretation. Furthermore, if we define quitative equivalence between two bound statesρ1 andρ2 by R(ρ1, ρ2) > 0 andR(ρ2, ρ1) > 0, we can fur-ther study classification of mixed states and compdifferent mixed states qualitatively.

As an initiative attempt to study entanglement echange rate, this work is preliminary and experimtal. Many related problems need further investigatisuch as bounding entanglement exchange rate, clating the exchange rate for some special statesanalyzing the exchange rate by defining different ty

-

of convertibility. Calculation of the exchange rate fany two entanglement will be a difficult task as othentanglement measures in the multipartite case. Hever, because of its definition, entanglement excharate is easily bounded. Such a rate or the lower bouof it tells how efficiently can the entanglement tranformation be carried out. We hope that the researcentanglement exchange rate can help us to undersand interpret various phenomena in quantum informtion theory.

Acknowledgement

We thank anonymous referee for helpful commeand suggestions.

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