An Entanglement Criterion for States in Infinite Dimensional Bipartite Quantum Systems

In this paper, an entanglement criterion for states in infinite dimensional bipartite quantum systems is presented. We generalize some of separability criterion that was recently introduced by Wu and Anandan in (Phys. Lett. A, 2002, 297, 4-8) to infinite dimensional bipartite quantum systems. In addition, we give an example aimed to illustrate the application of the theorem.


Introduction
Quantum entanglement plays a crucial role in the rapidly developing theory of quantum information and quantum computation [1].In the cases of finite dimensional quantum systems, there are many methods to quantify the entanglement of bipartite and multipartite quantum systems [2][3][4][5][6][7][8].However, most of them have not explicit formula, or it is hard to calculate.For the cases of infinite dimensional systems, the method of entanglement detection is a very difficult problem.But the case of infinite dimensional quantum systems can't be neglected since they do exist in quantum world [9].Recently, Shengjun Wu and Jeeva Anandan [10] proposed a necessary criterion based on Pauli matrices re-presentation.Their result is as follows: Let , where each kl   1, 2; 1, 2 k l   is an n by n matrix,  can also be written as: where the four matrices are n-dimensional Hermitian matrices.Let R be a 3-dimensional real matrix, and R  be a transformation on the density matrix  with the following form: where Shengjun Wu and Jeeva Anandan [10] give the following results: is positively defined for any vector where , , x y z M M M are defined in Equation (4).Then, a nature problem is arisen: whether or not there is counterpart result for the infinite-dimensional bipartite quantum systems?We find that the answer is "yes".The aim of the present paper is to establish this criterion for the infinite dimensional bipartite quantum systems.
The paper is organized as follows: In Section 2, we give the main results and the proof of the main results.In Section 3, we give an example to illustrate the application of the theorem.

Some Notations and Main Results
In this section, we mainly generalize the finite dimensional results, which be proposed by Wu and Anandan [10], to infinite-dimensional bipartite quantum sys- , where dim 2, dim .
H H be separable complex Hilbert spaces, by B , we denote the set of all states in . The set of all separable pure states in Throughout the paper we use the Dirac's symbols.The bra-ket notation stands for the inner product in the given Hilbert spaces.Recall that a quantum state positive and has trace one, is said to be separable if  can be written as: where Furthermore, it is shown in [11] that any separable state  admits a representation of the Bochner interal where  is a Borel probability measure on with respect to the trace norm.Where is the characteristic function of i , and with respect to the trace norm, as well as with respect to the Hilbert Schmidt norm.Where there exists an ensemble

 
, Next, we give the main results as follows: Theorem 2.1 Let H A and H B be separable complex Hilbert spaces, dim  2   R   must be positively defined for any 3 by 3 real matrix R which satisfies , where 0 is defined in Equations ( 5) and ( 6).
is positively defined for any vector are defined in Equation (4).
Proof. 1) Since  is separable, according by Equations ( 9)-( 12), we have where are pure states of B H , respectively.Furthermore, according by Bloch representation [1], we have with respect to the trace norm, where , , x y z  r are real vectors on the Bloch sphere and satisfies x y z    .Comparing Equations ( 3) and (15), we have . By E we denote the set of all partitions of where Since for any 3-dimensional real matrix R which satisfies , means that , on the other hand, according by Equation ( 16), we have It is obvious that we have , in fact since , so Since   a b a b , so we have and On the other hand, we have


This completes the proof.

Example
Next, we give an example to illustrate the application of the Theorem 2.1.We consider a bipartite infinite dimensional state with the following forms: where   0 , 1 is the orthogonal real basis of According by Theorem 2.1, by a straightforward calculation, we obtain the following result, if so  is entangled.
Remark: It is obvious that this criterion of Theorem 2.1 is weaker than PPT criterion [4], in fact if x ≠ 1,  is also entangled, but this criterion give us a method to detect the entanglement for states in infinite bipartite quantum systems.