Remarks on the Cross Norm Criterion for Separability

Remarks on the Cross Norm Criterion for Separability
Remarks on the Cross Norm Criterion for Separability

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Remarks on the Cross Norm Criterion for Separability S.J.Akhtarshenas a,b,c ?,M.A.Jafarizadeh a,b,c ?a Department of Theoretical Physics and Astrophysics,Tabriz University,Tabriz 51664,Iran.b Institute for Studies in Theoretical Physics and Mathematics,Tehran 19395-1795,Iran.c Research Institute for Fundamental Sciences,Tabriz 51664,Iran.February 1,2008

Abstract

Recently in Reference[quant-ph/0202121]a computational criterion of separability induced by greatest cross norm is proposed by Rudolph.There,Rudolph conjectured that the new criterion is not weaker than positive partial transpose criterion for separability.We show that there exist counterexample to this claim,that is,proposed criterion is weaker than the positive partial transpose criterion.

Keywords:Quantum entanglement,Bell decomposable states,Greatest cross norm,Positive partial transpose

PACs Index:03.65.Ud

1Introduction

Entanglement as the most non classical features of quantum mechanics has been attracted much attention in the past decade.Though,non local characters of quantum mechanics is singled out in many decades ago[1,2],but it has recently received considerable attention in connection with theory of quantum information[3,4,5].Entanglement is usually arise from quantum correlations between separated subsystems which can not be created by local actions on each subsystems.By de?nition,a bipartite mixed stateρis said to be separable if it can be expressed as

ρ= i w iρ(1)i?ρ(2)i,w i≥0, i w i=1,(1-1) whereρ(1)i andρ(2)i denote density matrices of subsystems1and2,respectively.Otherwise the state is entangled.

The central tasks of quantum information theory is to characterize and quantify entangled states. A?rst attempt in characterization of entangled states has been made by Peres and Horodecki family [6,7].It was shown that a necessary condition for separability of a two partite system is that its partial transpose be positive.Horodecki family show that this condition is su?cient for separability of composite systems only for dimensions2?2and2?3.

A new criterion for separability and also an entanglement measure for two partite systems based on greatest cross norm are introduced by Rudolph[8,9,10].In an interesting paper[10] he obtained the values of greatest cross norm for some states such as Werner states and isotropic states.In[10]Rudolph also introduced a computational criterion for separability of mixed states induced by greatest cross norm and he could obtain the separability conditions for some states such as Werner states,isotropic states and2-qubit Bell diagonal states.He showed that the new criterion completely characterizes separability properties of pure states,Bell decomposable states

and isotropic states in arbitrary dimension.He conjectured that new criterion is neither weaker nor stronger than positive partial transpose(PPT)criterion introduced by Peres and Horodeckis in[6,7].

In this paper we introduce Bell decomposable states of2?3systems and we show that there is state in this category that is entangled in the sense of PPT criterion but it is separable in the sense of new criterion introduced in[10],that is the new criterion is weaker than PPT criterion.

The paper is organized as follows.In section2we brie?y review greatest cross norm criterion for separability of two partite systems.The criterion induced by greatest cross norm is also reviewed. Bell decomposable states in2?3systems are introduced in section3and also PPT conditions for separability of these states is obtained.Finally we show that there exist state that is entangled in the sense of PPT criterion but satisfy new criterion for separability proposed by Rudolph.

2Trace class norm criterion for separability and associated in-duced separability criterion

In this section we brie?y review greatest cross norm for separability of two partite systems intro-duced by Rudolph in[8]and also the induced criterion introduced in[10].

Let us consider Hilbert spaces H1and H2associated with particles1and2,respectively.One can show that the spaces T(H1)and T(H2)of trace class operators on H1and H2are Banach spaces once they equipped with the trace class norm . (1)1and . (2)1,respectively.The algebraic tensor product T(H1)?alg T(H2)of T(H1)and T(H2)is de?ned as the set of all?nite sums n i=1u i?v i where u i∈T(H1)and v i∈T(H2)for all i.

A cross norm on T(H1)?alg T(H2)is de?ned by(see[8,10]and references therein)

t γ:=inf n i=1 u i 1 v i 1|t=n i=1u i?v i ,(2-2) where t∈T(H1)?alg T(H2)and the in?mum is taken over all?nite decompositions of t into

elementary tensors.The norm majorizes any subcross on T(H1)?alg T(H2)(a norm on T(H1)?alg T(H2)is subcross norm if t1?t2| ≤ t1 1 t2 1for all t1∈T(H1)and t2∈T(H2)and it is cross norm if saturates the inequality for all t1∈T(H1)and t2∈T(H2))is called greatest cross norm.

The greatest cross norm criterion proposed by Rudolph is de?ned as follows[8,10].Let H1

and H2be?nite dimensional Hilbert spaces andρbe a density operator on H1?H2.The density matrixρis separable if and only if ρ γ=1.Rudolph in[10]determines greatest cross norm for

some states such as Werner states and isotropic states.In addition in the second part of Ref.[10], Rudolph introduced a new necessary separability criterion for bipartite systems induced by the greatest cross norm on Hilbert-Schmidt space.

In the Hilbert-Schmidt space HS(H1?H2),the operators of the Hilbert space H1?H2are regarded as vectors.This space is equipped with the Hilbert-Schmidt inner product de?ned by

T|T′ =tr(T?T′),where T and T′are two operators acting on space H1?H2.Let HS(H1)and HS(H2)denote Hilbert-Schmidt spaces corresponding to Hilbert spaces H1and H2,respectively.It

has been shown in[10]that there exist a one-to-one correspondence between Hilbert-Schmidt oper-

ators T∈HS(H1?H2)and Hilbert-Schmidt operators U(T):HS(H1)?→HS(H1).Alternatively we can de?ne the trace class norm of U(T)denoted by T(U(T)).

Every state T∈HS(H1?H2)in the Hilbert-Schmidt space can be written as[10]

T= iλi E i?F i(2-3)

where{λi}i are non-negative real numbers and{E i}i and{F i}i are orthonormal bases of HS(H1) and HS(H2)respectively[10].Also the trace class norm of U(T)is equal to T(U(T))= iλi.

Rudolph in[10]proposed its new criterion for separability in a proposition which is quoted below:

Proposition1[10]Let H be a?nite dimensional Hilbert space andρ∈T(H?H)be a density operator.Ifρis separable then

T(U(ρ))≤1.(2-4) Based on the above criterion,Rodulph obtained separability conditions of some states such as Werner states,isotropic states and2-qubit Bell diagonal states.He conjectured that the new criterion is neither weaker nor stronger than the Peres-Horodeckies PPT criterion for separability. In the next section we present state that violates positive partial transpose criterion but satisfy the separability criterion given in Eq.(2-4).

3Bell decomposable states of2?3quantum systems

In this section we review Bell decomposable states of2?3quantum systems.A Bell decomposable density matrix acting on2?3Hibert state can be de?ned by

ρ=

6

i=1p i|ψi ψi|,0≤p i≤1,6 i=1p i=1,(3-5)

where|ψi are Bell states in H2?H3~=H6Hibert space,de?ned by:

|ψ1 =12(|11 +|22 ),|ψ2 =12(|11 ?|22 ),

|ψ3 =12(|12 +|23 ),|ψ4 =12(|12 ?|23 ),(3-6)

|ψ5 =12(|13 +|21 ),|ψ6 =12(|13 ?|21 ).

It is quite easy to see that the above states are orthogonal and thus span the Hilbert space of2?3 systems.

A necessary condition for separability is presented by Peres[6].He show that the matrix obtained from from partial transpose of separable state must be positive.Horodeckies[7]have shown that Peres criterion provides su?cient condition for separability only for composite quantum systems of dimension2?2and2?3.This implies that the state given in Eq.(3-5)is separable if and only if the following inequalities satisfy

(p1+p2)(p3+p4)≥(p5?p6)2,(3-7)

(p3+p4)(p5+p6)≥(p1?p2)2,(3-8)

(p5+p6)(p1+p2)≥(p3?p4)2.(3-9) On the other hand expanding Eq.(3-5)in terms of canonical base|i ?|j we get

ρ=1

U(ρ)=1

4 (p1?p2)2+(p3?p4)2+(p5?p6)2 ,

1

B=

((p1+p2)(p3+p4)+(p3+p4)(p5+p6)+(p5+p6)(p1+p2)).

4

It is easy to see that all eigenvalues are non-negative.Now we can easily determine the separability criterion given in Eq.(2-4)as

4

i=1 A+√B?C≤1.(3-14) In the rest of this section we shall present a counterexample to the claim that the criterion given in Eq.(2-4)is not weaker than PPT criterion for separability.Let us consider a Bell decomposable state given by

p1=0.3,p2=0,p3=0.2,p4=0.1,p5=0.4,p6=0.(3-15) It is quite easy to see that the state given by Eq.(3-15)violates PPT criterion given in Eq.(3-7), so it is entangled state.On the other hand,it is separable in the sense of criterion given in Eq.

(2-4).This implies that the new criterion induced by the greatest cross norm is weaker than the PPT criterion for separability.

4Conclusion

We have provided a counterexample to show that the newly proposed criterion of separability(in-duced by greatest cross norm)is proposed by Rudolph is weaker than the positive partial transpose criterion,therefore,we have still a long way ahead to solve the long standing separability criterion in mixed quantum states.

References

[1]A.Einstein,B.Podolsky and Rosen,Phys.Rev.47,777(1935).

[2]E.Schr¨Odinger,Naturwissenschaften.23807(1935).

[3]C.H.Bennett,and S.J.Wiesner,Phys.Rev.Lett.69,2881(1992).

[4]C.H.Bennett,G.Brassard,C.Cr′e peau,R.jozsa,A Peres and W.K.Wootters,

Phys.Rev.Lett.70,1895(1993).

[5]C.H.Bennett,D.P.DiVincenzo,J.A.Smolin and W.K.Wootters,Phys.Rev.A

54,3824(1996).

[6]A.Peres,Phys.Rev.Lett.771413(1996).

[7]M.Horodecki,P.Horodecki and R.Horodecki,Phys.Lett.A2231(1996).

[8]O.Rudolph,J.Phys.A333951(2000).

[9]O.Rudolph,J.Math Phys.422507(2001).

[10]O.Rudolph,eprint quant-ph/0202121.

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