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Quantum separability criteria for arbitrary-dimensional multipartite states

PHYSICAL REVIEW A89,022325(2014)

Ming Li,1,2Jing Wang,1,3Shao-Ming Fei,2,3and Xianqing Li-Jost2

1College of the Science,China University of Petroleum,266580Qingdao,People’s Republic of China

2Max-Planck Institute for Mathematics in the Sciences,D-04103Leipzig,Germany 3School of Mathematical Sciences,Capital Normal University,100048Beijing,People’s Republic of China

(Received29December2013;published18February2014)

We present separability criteria for both bipartite and multipartite quantum states.These criteria include the

criteria based on the correlation matrix and its generalized form as special cases.We show by detailed examples

that our criteria are more powerful than the positive partial transposition criterion,the realignment criterion,and

the criteria based on the correlation matrices.

DOI:10.1103/PhysRevA.89.022325PACS number(s):03.67.Mn,02.20.Hj,03.65.?w

I.INTRODUCTION

Quantum entanglement,as the remarkable nonlocal feature

of quantum mechanics,is recognized as a valuable resource

in the rapidly expanding?eld of quantum information sci-

ence,with various applications such as quantum computation

[1,2],quantum teleportation[3],dense coding[4],quantum

cryptographic schemes[5],quantum radar[6],entanglement

swapping[7],and remote state preparation(RSP)[8–11].

Quantum states without entanglement are called separable

states,which constitute a convex subset of all the quantum

states.Distinguishing quantum entangled states from the

separable ones is a basic and longer standing problem in the

theory of quantum entanglement.It has attracted great interest

in the last20years.

For mixed states we still have no general criterion.A

strong criterion,named partial positive transposition(PPT),

to recognize mixed entangled quantum state was proposed by

Peres in1996in[12].It says that for any bipartite separable

quantum states the density matrix must be semipositive under

partial transposition.Afterwards,by using the method of

positive maps the Horodecki et al.[13]showed that the

Peres’criterion is also suf?cient for2×2and2×3bipartite

systems.For high-dimensional states,the PPT criterion is only

necessary.Horodecki[14]has constructed some classes of

families of inseparable states with positive partial transposes

for3×3and2×4systems.States of this kind are said to be

bound entangled(BE).Another powerful operational criterion

for separability is the realignment criterion[15,16].It demon-

strates a remarkable ability in detecting the entanglement of

many bound entangled states and even genuinely tripartite

entanglement[17].Considerable efforts have been made in

proposing stronger variants and multipartite generalizations

for this criterion[18,19].It was shown that PPT criterion and

realignment criterion are equivalent to the permutations of the

density matrix’s indices[17].

Recently,some more elegant results for the separability

problem have been derived.In[20–22],a separability criteria

based on the local uncertainty relations(LUR)was obtained.

The authors show that for any separable stateρ∈H A?H B,

G A k?G B k

G A k?I?I?G B k

where G A

k or G B

are arbitrary local orthogonal and normalized

operators(LOOs)in H A?H B.This criterion is strictly stronger than the realignment criterion.Thus more bound entangled quantum states can be recognized by the LUR criterion.The criterion is optimized in[23]by choosing the optimal LOOs.The covariance matrix of a quantum state is also used to study separability in[24].It has been pointed out in[25]that the LUR criterion,including the optimized one,can be derived from the covariance matrix criterion.In [26]the author has given a criterion based on the correlation matrix of a state.The correlation matrix(CM)criterion is then shown to be independent of PPT and realignment criterion in[27],i.e.,there exist quantum states that can be recognized by the correlation criterion while the PPT, realignment criterion,and the covariance matrix criterion fail. In[28],by de?ning matricizations of the correlation tensors, the authors introduced a general framework for detecting genuine multipartite entanglement and nonfully separability in multipartite quantum systems.

In this paper,we present a generalized form of the correla-tion matrix criterion for bipartite quantum systems[26,27]and for multipartite quantum systems[29].Our criterion includes the criterion based on the correlation matrix as a special case and is more powerful than the later for detecting entanglement, as shown by detailed examples.Thus our criterion will be more ef?cient than the positive partial transposition criterion,the realignment criterion,and the covariance matrix criterion for some quantum states.

II.SEPARABILITY CRITERION FOR BIPARTITE

QUANTUM STATES

Let H d1

and H d2

be two vector spaces with dimensions d1and d2respectively.By using the generators of SU(d),λi, i=1,2,...,d2?1,any quantum stateρ∈H d1A?H d2B can be writing as

ρ=

d1d2

I?I+

k=1

r kλk?I+

l=1

s l I?λl

k=1

l=1

t klλk?λl,(1)

where r k=12d

Tr(ρλk?I),s l=12d

Tr(ρI?λl)and t kl= 1

Tr(ρλk?λl).We denote T the matrix with entries t kl and

LI,W ANG,FEI,AND LI-JOST PHYSICAL REVIEW A 89,022325(2014)

de?ne

?T =?

????

?1d 1d 2

s 1s 2···s d 22?1

r 1t 11t 12···t 1(d 22?1)r 2t 21t 22···t 2(d 22?1)···r d 21?1

t (d 21?1)1

t (d 21?1)2

···

t (d 21?1)(d 22?1)

??????

.(2)

Theorem 1.If ρ∈H d 1A ?H d 2

B is separable,then for any d 21?d 22matrix M and (d 21?1)?(d 22?1)matrix N with real entries m ij and n ij respectively,

m kl T kl

d 21?d 1+2 d 22?d 2+2 2d 1d 2σmax (M ),(3) kl

n kl t kl

(d 1?1)(d 2?1)4d 1d 2

σmax (N ),(4)

where σmax (M )and σmax (N )are the maximal singular values

of M and N ,respectively.

Proof.A separable quantum state ρcan be expressed as

ρ= i

p i |ψi ψi |?|φi φi |.(5)

By writing the pure states |ψi and |φi in their Bloch forms,

we have

ρ= i

p i |ψi ψi |?|φi φi |

p i

d 1

I +

x ik λk ?

1d 2

I +

y il λl

d 1d 2I ?I +1d 2 i

p i k

x ik λk ?I +1d 1 i

p i

l y il I ?λk + i p i kl

x ik y il λk ?λl .(6)Comparing (1)with (6),we have

r k =1d 2 i p i x ik ,s l =1d 1 i p i y il ,

t kl = i

p i kl

x ik y il .

(7)

De?ne ?x i =(1d 1,x i 1,...,x i (d 21?1))t and ?y i =(1d 2

,y i 1,...,y i (d 22?1))t

,where t stands for the transposition.Since |ψi ∈H d 1A and |φi ∈H d 2B are all pure states,one has

Tr(|ψi ψi |)2=Tr 1

d 1

I + k

x ik λk

1d 1

+2 k x 2ik =1,(8)i.e.,|| x i ||=

√

k x 2

ik =

d 1?1

2d 1

.Hence || ?x

i ||=

d 2

1?d 1+2

.Similarly we have || ?y

i ||=

d 2

2?d 2+2

2d 2

.Therefore for any real

matrices M and N ,one obtains

kl m kl T kl = ikl p i m kl ?x ik ?y il i

p i | ?x i ,M ?y i |

d 2

1?d 1+2 d 22?d 2+2

2d 1d 2

σmax (M );

kl n kl t kl = ikl p i n kl x ik y il i

p i | x i ,N y i |

(d 1?1)(d 2?1)4d 1d 2

σmax (N ).

The correlation matrix criterion in [26]illustrates that

if quantum state ρis separable,then the Key-Fan norm ||T ||KF √(d 1?1)(d 2?1)

4d 1d 2

.In the following we show the power of Theorem 1in detecting entanglement by two corollaries.Corollary 1.The criterion based on the correlation matrix is included in Theorem 1.

Proof.Let T =U V ?be the singular value decomposition of T .Since T is a real matrix,one can always choose U and V to be orthogonal matrices.Without loss of generality,we assume that d 1 d 2.Set N =(V U ?)t ,where is a block

matrix of the form (I 0)t ,I is the (d 21?1)×(d 2

1?1)identity

matrix,0stands for a (d 22?d 21)×(d 22?d 2

1)zero matrix.The singular values of N must be either 1or 0.One obtains

||T ||KF =|Tr(U V ?V U ?)|=|Tr(T N t

)|=

n kl t kl

(d 1?1)(d 2?1)4d 1d 2σmax (N )= (d 1?1)(d 2?1)4d 1d 2.

This means that one can get the correlation matrix criterion

from Theorem 1.

Corollary 2.If a bipartite quantum state ρ∈H d 1A ?H d 2

B is separable,then the following inequality must hold:

||?T

||KF d 2

1?d 1+2 d 22?d 2+2

2d 1d 2

,(9)where || ||KF =Tr √

?stands for the trace norm of .

Proof.Assume d 1 d 2.Let ?T

=X Y ?be the singular value decomposition of ?T

,with X and Y the corresponding orthogonal matrices.Set M =(Y X ?)t ,where =(I 0)t ,I

and 0are the d 21×d 21identity matrix and the (d 22?d 2

1)×(d 22?d 2

1)zero matrix,respectively.The singular values of M are either 1or 0.Then we obtain

||?T ||KF =|Tr(X Y ?Y X ?)|=|Tr(?T M t )|=

m kl ?T kl d 2

1?d 1+2 d 22?d 2+2 2d 1d 2

σmax (M )

= d 2

1?d 1+2 d 22?d 2+2 2d 1d 2,

which ends the proof of the corollary.

QUANTUM SEPARABILITY CRITERIA FOR ARBITRARY-...PHYSICAL REVIEW A 89,022325(2014)

Corollary 1shows that Theorem 1is not weaker than the correlation matrix criterion in detecting entanglement for

quantum states in H d 1A ?H d 2

B .In fact,by the following example we can show that Theorem 1is strictly stronger than the correlation matrix criterion,the realignment criterion,and the PPT criterion.

Example.A 3×3PPT entangled state is given in [30]

ρ=1

4 I 9?

4 i =0

|ψi ψi | ,(10)where |ψ0 =|0 (|0 ?|1 )/√2,|ψ1 =(|0 ?|1 )|2 /√

2,|ψ2 =|2 (|1 ?|2 )/√

2,|ψ3 =(|1 ?|2 )|0 /√2,and |ψ4 =(|0 +|1 +|2 )(|0 +|1 +|2 )/3.The state is shown to violate the correlation matrix criterion.Let us mix ρwith white noise:

σ(x )=xρ+

1?x

I 9.(11)

The correlation matrix criterion detects the entanglement for 0.9493

??????0.81340.1905?0.110.18?0.40670.17980000.19050.3849?0.243?0.8060.2608?0.0989000?0.11?0.2430.1043?0.3511?0.15060.87360000.1798?0.09890.8736?0.3258?0.1634?0.2898000?0.40670.2608?0.1506?0.1634?0.867?0.16340000.1798?0.806

?0.3511

?0.2898?0.1634

?0.32580000000000.964000

0000000.96400000

0.964

?????????????

,which has the maximal singular value 1.036.From (3)the state σ(x )is entangled for 0.94

III.SEPARABILITY CRITERION FOR MULTIPARTITE

QUANTUM STATES

In this section we consider the separability problem for N-partite quantum systems H 1?H 2?···?H N with dim H i =d i ,i =1,2,...,N .

Let λ{μk }

αk =I d 1?I d 2?···?λαk ?I d μk +1?···?I d N with λαk ,the generators of SU (d i ),appearing at the μk th position and

T {μ1μ2···μM }α1α2···αM = M

i =1d μi 2M N i =1d i Tr ρλ{μ1}α1λ{μ2}

α2···λ{μM }αM ,which can be viewed as the entries of the tensors T {μ1μ2···μM }

For αM =···=αN =0with 1 M N ,we de?ne that

?T α1α2···αN =T μ1···μM α1···αM ,and for α1=···=αN =0,de?ne that ?T α1···αN =1N k =1k

.Hence we have a tensor ?T with elements {?T

α1···αN ,αk =0,1,...,d 2k

?1}.If we set λ{k }0=I d k for any 1 k N ,then any multipar-tite state ρ∈H 1

?H 2?···?H N can be generally expressed

by the tensor ?T

as [29]ρ=

α1α2···αN

?T α1α2···αN λ{1}α1λ{2}α2···λ{N }αN

,(12)where the summation is taken for all αk =0,1,...,d 2

k ?1.To obtain the criterion for N -partite quantum systems,we adopt the de?nition of the n th matrix unfolding T n of a tensor T ,which is a matrix with i n to be the row index and the rest subscripts of T to be column indices (detailed description can

be found in Refs.[29,31]).The Ky Fan norm of the tensor T over N matrix unfoldings is de?ned as

||T ||KF =max {||T n ||KF },n =1,2,...,N.

(13)

Theorem 2.If a quantum state ρ∈H 1?H 2?···?H N is fully separable,then for any tensors M and W with real entries

m i 1i 2···i N ,i k =1,2,...,d 2k ?1,and w j 1j 2···j N ,j l =1,2,...,d 2

k ,we have

i 1i 2···i N

m i 1i 2···i N T i 1i 2···i N N k =1 d k ?12d k σmax (M ),(14) j 1j 2

···j N

w j 1j 2···j N ?T i 1i 2···i N N k =1 d 2k ?d k +22d 2

σmax (W ),(15)

where σmax (M )and σmax (W )stand for the maximal eigenvalue

of the matrix unfolding M n and W n .The maximum is taken over all kinds of mode n matricization.

Proof.Assume that ρ∈H 1?H 2?···?H N is fully separable,one can always ?nd the following decomposition:

ρ= i

p i ψ1i ψ1i ? ψ2i ψ2i ?···? ψN i ψN i

(16)where |ψm i ψm i |are density matrices of pure states in H m .Using the Bloch representation of density matrix,we have

ψm i ψm i =1d m

I + α

x m iαm λαm

,(17)where x m

iα

m =Tr(|ψm i ψm i |λαm )/2.By (8)one has that || x m i ||=√d m ?1

2d m .Denote ?x m i =(1d m

,x m i 1,...,x m i (d 21?1))t .We ob-tain that || ?x m i ||= d 2

m ?d m +22

.Substituting (17)into (16)one

LI,W ANG,FEI,AND LI-JOST PHYSICAL REVIEW A 89,022325(2014)

has that ρ=

1 N k =1d k ?N k =1I k +

μ1α1

d μ1

N k =1 i p i x μ1iα1λμ1α1

μ1μ2α1α2

d μ1d μ2 N

k =1 i

p i x μ1iα1x μ2iα2λμ1α1λμ2

α2+···+

μ1···μM ,α1···αM

M k =1d μk N k =1

p i x μ1iα1···x μM iαM λμ1α1

···λμM

αM +

α1···αN

p i x 1iα1···x N iαN λ1α1

···λN αN .(18)

Comparing (12)and (18),one gets

T {μ1μ2···μM }α1α2···αM

= M k =1d μk N k =1

p i x μ1iα1···x μM iαM

.(19)

According to the de?nitions of x m i , ?x

m i ,and T α1α2···αN ,?T α1α2···αN ,we have that

T α1α2···αN = i

p i x 1iα1···x N

iαN =

p i x 1i ? x 2i ?···? x N i ,

(20)

α1α2···αN =

p i ?x 1iα1···?x N

iαN

p i ?x

1i ? ?x 2i ?···? ?x N i ,(21)

where ?stands for the out product.

Let M n be mode n matricization of M .Then for any tensor M we have that

i 1i 2···i N

m i 1i 2···i N T i 1i 2···i N

p i x n i ,M n x 1i ?···? x ?n i ?···? x N i t N k =1

d k ?1

2d k

σmax (M ).Inequality (15)can be derived similarly.

In [29],the authors have derived a generalized form of the correlation matrix criterion which says that if a quantum state ρ∈H 1?H 2?···?H N is fully separable,then

||T ||KF =||T n ||KF N k =1

d k ?1

2d k .(22)Here we show that one can obtain the generalized correla-tion matrix criterion from Theorem 2.

Corollary 3.Inequality (22)is included in theorem 2.Moreover,if quantum state ρ∈H 1?H 2?···?H N is fully separable,then the following inequality holds:

||?T ||KF =||?T n ||KF N k =1

d 2k ?d k +22d k .(23)Proof.Assum

e that the n th unfold T n is just the one to attain the ||T ||KF .One immediately derives a singular value

decomposition of T n ,T n =V n n U ?

n for some orthogonal matrices V n and U n .Let M be the tensor with the n th

matrix unfolding M n =V n n U ?

n ,where n =(I 0),I is the

(d 2n ?1)×(d 2n ?1)identity matrix and 0is the zero matrix

with order such that n is a (d 2

?1)× N

k =1(d 2k

?1)

(d 2n

?1)matrix.Since both V n and U n are orthogonal matrices,the maximal singular value must be 1.From Theorem 2we have

i 1i 2···i N

m i 1i 2···i N T i 1i 2···i N =Tr(M n T ?n )=Tr(V n n U ?n U n n V ?

n )=Tr( n )

=||T ||KF N k =1

d k ?1

2d k

,which leads to the inequality (22).Inequality (23)can be proved similarly. Corollary 3can detect some PPT entangled quantum states in multipartite quantum systems,such as the three-qutrit bound entangled states ρc ?|ψ ψ|considered by Clarisse and Wocjan [32],where

ρc =112?

???????

?10100010001000?10?101020?1000000010?10100

0?10101000?10?1020001000102000?10100020000000000??

???

??????is the chess-board state and |ψ is an uncorrelated ancilla.If

we mix ρc ?|ψ ψ|with white noise and de?ne σ=pρc ?

|ψ ψ|+1?p

27I ,the entanglement is detected for 0.83265

IV .CONCLUSIONS AND REMARKS

It is a basic and fundamental question to distinguish separable quantum states from entangled ones.Although the quantum separability problem has been shown to be NP hard,it is possible to derive some necessary criteria of separability.We have derived separability criteria of quantum states for both bipartite and multipartite quantum ones.The criteria are shown to be more ef?cient in detecting quantum entanglement of some quantum states than the (generalized)criterion based on the correlation matrix,the PPT criterion,the realignment criterion,and the covariance matrix criterion.Similar to the case of previous separability criteria,our criteria can also be used to derive lower bounds for concurrence.

ACKNOWLEDGMENTS

This work is supported by the NSFC Grants No.

11105226and No.11275131;the Fundamental Research Funds for the Central Universities,Grants No.12CX04079A and No.24720122013;and Research Award Fund for out-standing young scientists of Shandong Province,Grant No.BS2012DX045.

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