Sufficient and necessary conditions of separability for bipartite pure states in infinite-dimensional systems

The PHC criterion and the realignment criterion for pure states in infinite-dimensional bipartite quantum systems are given. Furthermore, several equivalent conditions for pure states to be separable are generalized to infinite-dimensional systems.

In quantum mechanics, the states of the system are described by density operators which are positive linear operators on a complex Hilbert H with unit trace and the observables are represented by self-adjoint operators on H. A density operator that is a projection on a single vector is referred to as representing a pure state, while other density operators are representing mixed states.
Let H and K be complex separable Hilbert spaces and the tensor product Hilbert space H⊗K be the corresponding bipartite quantum system. By S(H), S(K) and S=S(H⊗K), we denote the sets of states on H, K and H⊗K, respectively. Recalling that, a pure state S ρ ∈ is called separable if, as the same to the finite-dimensional systems,  (1) or can be approximated in the trace norm by the states of the above form [1,2], where the series converges in trace norm, ρ (1) , ρ i (1) and ρ (2) , ρ i The quantum entangled states have been used as a basic resource in quantum information processing and communication [3,4], such as quantum cryptography [5,6], quantum computation [7,8], quantum secret sharing [9,10], quantum dense coding [11] and so on. Recently, entanglement in different models are investigated, for instance, the Jaynes-Cummings model [12], the Heisenberg XXZ model [13,14], the QED model [15] and the CV photon model [16]. To decide whether or not a state of bipartite quantum system is entangled is one of the most challenging tasks of this field [3,17]. For the finite-dimensional case, several entanglement criteria are proposed, such as the "positive partial transposition (PPT) criterion" [18,19], the "partial Hermitian conjugate (PHC) criterion" (a criterion for pure states [20]) and the 'realignment criterion' [21][22][23][24] For a density operator ρ, we denote by

Partial Hermitian conjugate criterion for pure states
In this section, we introduce the concept of partial Hermitian conjugate (PHC) of pure states in infinite-dimensional bipartite systems, and generalize the PHC criterion in [20] to infinite-dimensional case.
At first, we give the definition of PHC for separable pure Then, similar to the finite-dimensional case, the partial Hermitian conjugate of ρ is defined by PHC , , , However, for a general bipartite pure state, it is uneasy to calculate its partial Hermitian conjugate since it also needs complex conjugation on the corresponding coefficients [20]. To avoid this difficulty, we consider the so-called Schmidt decomposition of a given unit vector in H⊗K. Notice that, similar to the finite-dimensional case, each vector in infinitedimensional tensor product Hilbert space of two Hilbert spaces has a Schmidt decomposition [25]. In [25] Note that k and k′ are orthonormal sets since U and V are unitary operators. It is clear that Remark that, for any given pure state ρ ψ ψ = , as the Schmidt decomposition of ψ is uniquely determined, ρ PHC is uniquely determined and thus the PHC operation for pure states is well defined.
We are now ready to prove the PHC criterion for the infinite-dimensional case.
Proof. If ρ ψ ψ = is a separable pure state and , , , It follows that PHC , , , Note that It remains to show that if ρ PHC =ρ, then ρ is separable. If ρ PHC =ρ, we assume that the Schmidt decomposition of ψ Notice that the above equation is impossible whenever k l ≠ and it holds if and only if k=l=1. This leads to   Together with eq. (3), we see that the PHC operation for ρ α is not well defined. Remark 1.6. Just as pointed out in [20], the PHC criterion does not provide us with much convenience, and it only leads us to consider the entanglement problem from a different point of view. In addition, another application of this criterion may be that it can provide a new way to quantify the amount of entanglement of a given state by calculating some distance (such as trace distance) between the given state and the partial hermitian conjugate of the state. However, by Remark 1.5, it can only be used to pure states.

Realignment criterion for pure states
For the finite-dimensional case, the realignment criterion [22][23][24] is a practical criterion that can detect some PPT entangled states [23]. The aim of the present section is to generalize this criterion to infinite-dimensional bipartite quantum systems. We recall some known facts about the realignment criterion for the finite-dimensional systems. Assume dimH=n<∞ and dimK=m<∞ for a moment. In A similar argument as that for finite-dimensional case [21] shows that the realignment operation is well defined.
The theorem below is the main result of this section which generalizes Proposition 1 in [21] to infinitedimensional case. Moreover, Proof. If The relation (5) is obvious. Remark 2.3. The realignment criterion is valid for mixed states in infinite-dimensional systems, too. In fact, it is shown in [27] that, for any state Theorem 2.2 provides a practical tool to determine whether or not a given pure state is separable since the trace norm of the realignment operator can be calculated easily. In addition, the realignment criterion also implies that the

Some sufficient and necessary conditions of separability for pure states
So far, there are few practical criteria that are both necessary and sufficient for mixed states to be separable [28]. However the situation for pure states is much better. In the previous sections we have proposed two entanglement criteria for pure states in infinite-dimensional systems. In the present section we will give some more criteria of the separability of pure states. The main result is the following. Proof.
(2) ⇒ (4). For any j, k, let If ω ω is PPT, then, by eq. (6), for any j, k with j<k, Therefore there exists a contractive operator V jk such that for all j, the above equation yields It follows that k x and j x are linearly dependent.
As 0 ω ≠ , so    ( 3 ) ⇒ ( 4 ) . Therefore D ω is of rank one and thus (4) holds. ■ Next, we give a proposition related to the item (3) of Theorem 3.1. Before doing this we recall some known facts. In [28,29], the so-called entanglement witness criterion was

Proof.
Let By calculating, we can obtain that (1)  In conclusion, we have given several sufficient and necessary conditions of separability for pure states in infinite-dimensional bipartite systems which provide some principal insights into the structure of the pure states. Some conditions provide tools to study the entanglement measures on the set of pure states.