Quantum entanglement is a fascinating phenomenon in quantum physics. In recent decades, much works have been devoted to understand entanglement as it plays important roles in many quantum information processing. Nevertheless, there are still many problems remain unsolved in the theory of quantum entanglement. One basic problem is to determine whether a given bipartite state is entangled or separable. Although the problem is believed to be a nondeterministic polynomial-time hard problem, there are a number of operational criteria to deal with the problem, for example, the positive partial transpose (PPT) criterion,^[1,2] realignment criteria,^[3–7] covariance matrix criteria,^[8–10] correlation matrix criteria,^[11–13] and so on. More recently, some more separability criteria have been proposed.^[14–19] Among them, Li et al.^[14] presented separability criteria based on correlation matrices and the Bloch vectors of reduced density matrices. By adding some extra parameters, reference [19] present a more general separability criterion for bipartite states in terms of the Bloch representation of density matrices.

The state of two quantum systems A and B, acting on the finite-dimensional Hilbert space , is described by the density operator ρ. A state ρ is said to be separable if ρ can be written as a convex combination of product vectors,^[20] i.e., 1 where , , and ( and ). The state ρ is said to be entangled, when ρ cannot be written as in form of Eq. (1).

In this article, we put forward a new Bloch representation in terms of the Heisenberg–Weyl (HW) observable basis.^[21] It is one of the standard Hermitian generalization of Pauli operators, constructed from HW operators.^[22–25] They have distinct properties from those of Gell–Mann matrices,^[21] Based on the Heisenberg–Weyl representation of density matrices, we give a new separability criterion for bipartite quantum states and multipartite states. By example, we show that this criterion has advantages in determining whether a quantum state is separable or entangled.

HW observable basis First, we briefly introduce the HW-operator basis.^[21] The generalized Pauli “phase” and “shift” operators are given by and , respectively, mod and . Q and P are the discrete position and momentum operators describing a d × d grid.

The phase-space displacement operators for d-level systems are defined by 2 i.e.,^[26] 3 . These non-Hermitian orthogonal basis operators satisfy the following orthogonality condition, . The complete set of Hermitian operators can be constructed from the HW operators by defining 4 where . are the so-called HW observable basis and satisfy the orthogonality condition, 5 This basis simply reduces to the pauli matrices for d = 2. When d = 3, we have

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A state of single quantum system can be expressed in terms of the d × d identity operator I_d and the traceless Hermitian HW observable operators , 6 where . The coefficients r_lm in Eq. (6) are given by

Proof According to Eq. (6), we have Since ρ is a pure state, one has . Therefore .

Now consider bipartite states . Any state ρ can be similarly represented as^[27] 9 In particular, where , , and .

Similar to (7), we denote , , , and , , , , where t stands for transpose. Set , where the entries t_ij are given by the coefficients t_lmkn, , , , with the first two indices lm associated with the array index i, and the last two indices kn with the column index j of T.

Let us consider the following matrix: 10 where α and β are nonnegative real numbers, m is a given natural number, is an m × m matrix with all entries being 1. We have the following theorem.

Proof Since ρ is separable, from Ref. [13] there exist vectors , satisfying Eq. (8), and weights p_i satisfying , such that From Lemma 1, we have The matrix (10) has the form, Hence,

Accounting to that for any vectors and , one has 12 we have which gives rise to (11).

For high-dimensional quantum states, let us consider the following 2 × 4 bound entangled state^[28] as an example: where . We mix the above state with state , By choosing Theorem 1 can detect the entanglement in for , while Theorem 1 in ^[19], the V–B criterion^[13] and the L–B criterion^[14] can only detect the entanglement in for , , and , respectively. In this case, our criterion is better in detecting entanglement.

Here, instead of (4) if we define , , then , and the conclusion becomes In this case, the least upper bound in Theorem 1 is equal to the Theorem 1.^[19]

We now generalize our result in Theorem 1 to multipartite case. Let be an tensor, A and be two nonempty subsets of satisfying . Let denote the A, matricization of , see Ref. [19] and Ref. [29] for detail.

For any state ρ in , we import a natural number m and nonnegative real parameters , and define where . Let be the traceless Hermitian HW observable basis and satisfy the orthogonality relation . Denote the tensor given by elements of the following form: where and . Below we give the full separability criterion based on .

Proof Without loss of generality, we assume Since ρ is fully separable, from Ref. [30] there exist vectors such that where . Thus where we have used the equality (12) and .

As for tripartite case, by taking m=1, and , we consider the one-parameter three-qutrit state^[31] , with . Theorem 2 can detect the entanglement in ρ for , while Theorem 2 in Ref. [19] can only detect the entanglement in ρ for .

We have studied the separability problem based on the Bloch representation of density matrices in terms of the Heisenberg–Weyl observable basis. New separability criteria have been derived for both bipartite and multipartite quantum systems, which provide more efficient ways in detecting quantum entanglement for certain kinds of quantum states. These criteria can experimentally implemented.

In Ref. [19] the traceless Hermitian generators of SU(d) satisfying the orthogonality relations have been used in the Bloch representation of density matrices. While in this paper we have adopted the same approach as the one used in Ref. [19] but used the Heisenberg–Weyl observable basis^[21] in the Bloch representation of density matrices. An interesting fact here is that the ability of detecting quantum entanglement can be improved by using different observable basis in the Bloch representation. Hence our results are complementary to the ones obtained in Ref. [19] in detecting entanglement for some quantum states. Just like the case that one needs different witnesses to detect the entanglement of different quantum states, we need to measure the quantum systems with suitable local observable sets in entanglement detection. The choices of suitable observable basis depend on the detailed entangled states to be detected. It would be more interesting if such state-dependent choices of observable basis can be analytically derived optimally. It is also possible to improve such separability criteria by taking into account measurement outcomes of more observable bases simultaneously, similar to the cases that involve all the mutually unbiased bases,^[32,33] or mutually unbiased measurements,^[34] or general symmetric informationally complete positive operator-valued measurements.^[35]