Separability Criteria Based on the Weyl Operators.

Entanglement as a vital resource for information processing can be described by special properties of the quantum state. Using the well-known Weyl basis we propose a new Bloch decomposition of the quantum state and study its separability problem. This decomposition enables us to find an alternative characterization of the separability based on the correlation matrix. We show that the criterion is effective in detecting entanglement for the isotropic states, Bell-diagonal states and some PPT entangled states. We also use the Weyl operators to construct an detecting operator for quantum teleportation.

1. Introduction

Quantum information processing is responsible for implementing tasks such as super-dense coding [1], teleportation [2] and key generation [3]. Quantum entanglement is one of the key sources [4] of quantum advantages and has many applications ranging from quantum teleportation to quantum cryptography [5]. In recent years, much effort has been devoted to understanding entanglement, but still many problems remain unsolved. One key problem is to determine whether a given bipartite state is entangled or not. Recall that a bipartite quantum state in a Hilbert space is separable if where is a probability distribution, and are the reduced density matrices of subsystem and respectively. Otherwise is said to be entangled.

For low-dimensional bipartite systems such as , and systems, the celebrated PPT criterion [6] is a necessary and sufficient condition for separability. However, for higher-dimensional multipartite systems, entanglement detection is widely believed to be an NP-hard problem. Nevertheless there are several separability criteria available. Among them, a notable one is entanglement witness which detects entanglement theoretically and experimentally [7], and most linear separability criteria can be regarded as entanglement witnesses. As a nonlinear separability criterion the local uncertainty relation [8] is an effective method to detect entanglement, and there are some nonlinear criteria based on matrix methods, for example, the realignment criterion [9], the covariance matrix criterion [10] and the separability criteria based on the correlation matrix [11,12].

In this paper, we focus on an improved Bloch representation of density matrix in terms of the Weyl basis to derive separability criteria. Our work shows that the Weyl basis is advantageous in handling higher dimensional quantum states as well as revealing the symmetry property. This new scheme markedly simplifies calculations involving density matrices. Our method further demonstrates that Weyl operators can be widely applied in quantum information realm [13,14,15]. Particularly, Weyl operators also play important roles in entanglement detection [16,17,18,19,20,21,22,23]. In fact, Ref. [16] provided the generalized Pauli matrices based on the Weyl operators, and proposed a criterion to detect entanglement by the bounds of the sum of expectation values of any set of anti-commuting observables. Moreover, the separability criteria in terms of the Weyl operators for bipartite and multipartite quantum systems were presented in Ref. [17]. The Weyl operators also play a crucial role in constructing the Weyl discrete channels. Furthermore, the Weyl operators have been widely used in representation theory of affine Lie algebras and Yangians [24].

The layout of the paper is as follows. In Section 2 we first show that Weyl operators provide a generalization of the Pauli operators that can represent any quantum state in a tensor format. Based on the Weyl representation of quantum state, we will present a separability criterion in terms of the correlation matrix for an arbitrary bipartite quantum state in Section 3. Our method also gives a necessary and sufficient condition for separability, which is applicable in quantum teleportation, as shown in Section 4. Detailed examples are provided to illustrate the advantages of this method compared with previous methods.

2. The Representation of Quantum States in Terms of Weyl Operators

Let be a d-dimensional Hilbert space with computational basis , and denotes the finite field of modulo d integers. For simplicity all integers in the subscripts are modulo d. Recall that the Weyl operators are defined by Clearly the set forms a basis of linear generators in the general linear Lie algebra . We remark that the Weyl basis is also called the principal basis in the literature. When , the Weyl operators specialize to the Pauli matrices, i.e., . In general when , the Weyl basis is different from both the Cartan–Weyl basis and Gell-Mann basis. The Weyl operators enjoy the following algebraic relations Although the Weyl operators are not Hermitian in general, they are unitary and enjoy the orthogonal relation where is the Kronecker symbol. Subsequently the Weyl operators obey the trace relation

As an example, there are nine linearly independent Weyl operators on a 3-dimensional Hilbert space listed as follows: where is a 3rd primitive unit root of 1, i.e., .

Since the linearly independent Weyl operators form a basis of , every density matrix can be uniquely expressed as a linear combination of the Weyl basis: where the coefficients for . Since , the coefficients satisfy the symmetry condition where * means complex conjugation. We also call the Bloch vector of relative to the Weyl basis and its length is defined as . Therefore any density matrix in Hilbert space can be uniquely characterized by a dimensional vector with the symmetry condition (4), that is, real parameters.

For any d-dimensional quantum state ρ in the form of Equation ( In particular, the equality holds if and only if ρ is pure.

Since any quantum state satisfies the trace condition , one obtains that Note that and the matrices are traceless for , the only nonzero terms in the summation are for and , that is, Using the symmetry condition (4), the trace is simplified as Therefore . If is pure, we have , which means . This completes the proof. □

Theorem 1 tells us that all Bloch vectors lie within a hypersphere of radius with the pure states on the spherical surface. Moreover, the quantum state is determined by the Bloch vector satisfying with the symmetry condition (4), thus the set of the Bloch vectors is a subset of the vector space with parameters.

3. Application of Weyl Operators in Separability Criteria

With the Weyl basis, a quantum state in space with and can be decomposed as where the coefficients , , , and are the identity operator and the Weyl operators of space , respectively. Let and be two complex vectors with dimension and , respectively, that is, where t denotes transposition. The entries form a matrix M with size , which will be referred to as the correlation matrix of (relative to the Weyl bases). It is the correlation matrix of under the Weyl basis. Since , the entries satisfy the symmetry conditions: , , .

According to the decomposition in Equation (8), the reduced states of on the two subsystems are respectively given by

A bipartite quantum state ρ in the form of Equation ( for some column vectors α and β.

One notices that Equation (8) can be rewritten as Since the matrices are linearly independent, if and only if for , that is, , and this completes the proof. □

Since any mixed state is a convex combination of pure states, Theorem 2 provides a necessary condition for separability for any mixed state in a bipartite system.

Now we denote the Ky Fan matrix norm of M as , which is the sum of the singular values of the matrix M. Notice that the Ky Fan norm, the trace norm and the Shatten-1 norm are uniform for a square matrix. Then we have the following necessary condition for separability for any bipartite quantum state.

If a bipartite state ρ in the form of Equation (

Suppose the quantum state is separable, then there exists a series of , , such that , with and . Suppose and with Bloch vectors and , respectively. Therefore the correlation matrix M of is . One sees that where we have used the equality of norm [25], for any pure state and , This completes the proof. □

Consider the isotropic state Therefore, the isotropic state can be represented as Note that the Ky Fan norm of correlation matrix M of

Let ρ be the following where

The Bell-diagonal states can be represented as Then the Ky Fan norm of the correlation matrix is

Consider the bipartite state where Therefore, the Ky Fan norm of the correlation matrix M of state

4. Application of Weyl Operators in Quantum Teleportation

In the process of quantum teleportation, the optimal fidelity of teleportation as an entangled resource can be expressed by fully entangled fraction [29,30,31]. For a given quantum state in d-dimensional Hilbert space, the optimal fidelity of telepotation with respective to can be described by the function where is the fully entangled fraction with respect to defined by [30]: where U runs through all unitary matrices, I is the identity matrix, and is the maximally entangled state. A state is a useful resource for teleportation if and only if [30]. If , the fidelity is considered to be no better than separability. In this sense, the fully entangled fraction can be used to detect a quantum teleportation resource. Ref. [32] gave an elegant formula for a two-qubit system by using the method of Lagrange multipliers. Refs. [33,34] constructed the teleportation witness for detecting the quantum states that are useful for quantum teleportation.

Now we construct an operator using the Weyl representation to detect if a quantum state is useful for quantum teleportation. Since the maximally entangled state can be decomposed as Equation (16) according to the Weyl operators [35], we let and define a normal operator by We claim that the operator can be used to detect whether an unknown quantum state is available for teleportation.

The quantum state ρ in a quantum system

For any quantum state , it has

Since , and quantum state is useful for quantum teleportation if and only if . Note that the maximum value is attainable since SU(d) is compact. Therefore the quantum state is useful for quantum teleportation if and only if there exists some unitary U such that . This completes the proof. □

Consider the following bipartite state [ where

5. Conclusions

We have investigated the Bloch decomposition of a density matrix relative to the Weyl basis in the quantum system. The geometric properties of Bloch vectors, including the length, are described in detail. For the bipartite quantum states, we have provided a necessary condition of separability in terms of the Ky Fan norm of the correlation matrix. Furthermore, we have demonstrated the feasibility and effectiveness of our separability criterion in detecting entanglement using examples of isotropic states, Bell-diagonal states and some PPT entangled states. Finally, we have constructed an operator based on the Weyl operators for detecting useful resources for quantum teleportation.