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International Journal of Modern Physics B

A simple entanglement criterion of two-qubit system

Chao Zhao, Qizhi Guo, Weihan Tan

Published Jul 20, 2019• 3 citations
Publisher

Abstract

We present a novel method to judge the entanglement of two-qubit system, the density matrix of a two-qubit system can be constituted by the principle density matrix [Formula: see text] and the separable density matrices [Formula: see text]–[Formula: see text]. The necessary and sufficient conditions for the two-qubit system, the three involved coefficients p [Formula: see text] 0, p [Formula: see text] 0, p [Formula: see text] 0, and the principal density matrix [Formula: see text] being separable, then the two-qubit system is separable, otherwise the two-qubit system is entangled. Finally, the criterion for several known density matrices have been verified in this way.

DOI
10.1142/s0217979219501972

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We present a novel method to judge the entanglement of two-qubit system, the density matrix of a two-qubit system can be constituted by the principle density matrix [Formula: see text] and the separable density matrices [Formula: see text]–[Formula: see text]. The necessary and sufficient conditions for the two-qubit system, the three involved coefficients p [Formula: see text] 0, p [Formula: see text] 0, p [Formula: see text] 0, and the principal density matrix [Formula: see text] being separable, then the two-qubit system is separable, otherwise the two-qubit system is entangled. Finally, the criterion for several known density matrices have been verified in this way.
The “separability problem” in quantum information theory is a quite important and well-known hard problem. The low-dimensional system satisfies the PPT criterion. However, the high-dimensional system problem has been shown to be NP-hard problem. In general, it is very difficult to find the analytic solution of the density matrix for the high-dimensional system. Therefore, getting an analytic solution for two-qubit system is an interesting and useful problem. We propose a novel criterion for separability and entanglement-verification of two-qubit system. We expressed the density matrix by a sum of a principal density matrix and six separable density matrices. The necessary and sufficient conditions for the two-qubit system include that if the four involved coefficients [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] and the principal density matrix [Formula: see text] are separable, then the two-qubit system is separable, otherwise the two-qubit system is entangled. Finally, our criterion results in a totally different conclusion compared to Horodecki’s criterion. We believe that the new criterion is more stringent than existing PPT methods.
In previous paper ( 2019 Int. J. Mod. Phys. B 33 1950197 ; 2020 Int. J. Mod. Phys. B 34 2050022 ), we presented a method to judge the entanglement of 2-qubit system. The necessary and sufficient conditions for the 2qubit system being separable are that if the relevant coefficients is positive and the principal density matrix is separable, then the system is separable, otherwise it is entangled. Now in this paper, we try to generalize this criterion to a 3-qubit system, and then, we further generalize the criterion of 3-qubit system to an N-qubit system. This is a complicated and interesting issue.
Abstract The determination of quantum state separability is the basic problem of quantum entanglement theory. By judging the rank of the matrix vector group formed by the density matrix block, the necessary and sufficient conditions for the separation of quantum pure States are given, and the necessary conditions for the separation of quantum mixed States are given, too. In the composite system 2 × 2, the rank of the vector group corresponding to the divisible pure state is 1, and the rank corresponding to the entangled state is 4.
In this Letter we find the new criteria of separability of multipartite qubit density matrixes. Especially, we discuss in detail the criteria of separability for tripartite qubit density matrixes. We find the sufficient and necessary conditions of separability for tripartite qubit density matrixes, and give two corollaries. The second corollary can be taken as the criterion of existence of entanglement for tripartite qubit density matrixes. In concrete application, its steps are quite simple and easy to operate. Some examples, discussions and the generalization to more high dimensional multipartite qubit density matrixes are given.
It is known that realignment crierion is necessary but not a sufficient criterion for lower as well as higher dimensional system. In this work, we first consider a two-qubit system and derived the necessary and sufficient condition based on realignment operation for a particular class of two-qubit system. Thus we solved the problem of if and only if condition partially for a particular class of two-qubit state. We have shown that the derived necessary and sufficient condition detects two-qubit entangled states, which are not detected by the realignment criterion. Next, we discuss the higher dimensional system and obtained the necessary condition on the minimum singular value of the realigned matrix of d⊗dd\otimes dd⊗d dimensional separable states. Moreover, we provide the geometrical interpretation of the derived separability criterion for d⊗dd\otimes dd⊗d dimensional system. Furthermore, we show that our criterion may also detect bound entangled state. The entanglement detection criterion studied here is beneficial in the sense that it requires to calculate only minimum singular value of the realigned matrix while on the other hand realignment criterion requires all singular values of the realigned matrix. Thus, our criterion has computational advantage over the realignment criterion.
Explicit separable density matrices, for mixed two qubits states, are derived by the use of Hilbert Schmidt decompositions and Peres Horodecki criterion. A strongly separable two qubits mixed state is defined by multiplications of two density matrices given with pure states while weakly separable two qubits state is defined by multiplications of two density matrices which includes non-pure states. We find the sufficient and necessary condition for separability of two-qubits density matrices and show that under this condition the two-qubit density matrices are strongly separable.