Tripartite Separability And Coherence Of W And Dicke Mixed State In Noise Environment

Quantum entanglement is an essential physical resource.It represents the non-classical and non-local correlation between multiple quantum systems,and has a wide range of applications in quantum communication.However,quantum entangled states are easily destroyed by noise（the simplest is white noise）and become separable in the process of use.Therefore,finding the minimal white noise tolerance for a particular entangled state becomes a highly relevant topic.In addition,for the study of the separability of bipartite entanglement,various entanglement detection approaches have been proposed,such as the positive partial transpose（PPT）criterion and so on.But for multipartite entanglement which is defined as the entanglement of more than two quantum systems,the situation is far more intricate,and except the full separability and the full entanglement,there are also partial separability,so the separability of multipartite quantum systems is still a problem worthy of further study.W state and Dicke state are two common quantum states,in this paper,we will study the tripartite separability of a four-qubit W and Dicke state in noise environment,as an example of seeking possible solution for the partial separability of multipartite entanglement systems.Quantum coherence originates from the superposition of quantum states,it reflects the ability of quantum states to exhibit quantum-interference effect.How to quantify the coherence of a specific state is one of the most critical issues in coherence research.In this paper,the robustness of coherence and ₁l-norm coherence measurement function are used to quantify coherence of states at the tripartite separability boundary of the four-qubit W and Dicke state in noise environment,and the results are compared and verified.The main content of this paper is the tripartite separability and quantifying coherence of a four-qubit W and Dicke state in noise environment.The first part is the study of tripartite separability.Based on entanglement witnesses and the method of constructing separable quantum states,the necessary and sufficient conditions near the W state are obtained,and the analytical formula is given.In addition,a necessary condition is obtained by using the numerical calculation combined with Wootters formula near the Dicke state,and the sufficient condition is obtained through the approach of constructing the quantum state with numerical calculation.The necessary criterion coincides with the sufficient criterion and agrees with the result of numeric results.The second part is the study of quantifying coherence,the robustness of coherence can be reconstructed as the expectation of the coherent witness operator for any quantum state.According to the characteristic coefficient of the state and the witness operator,the expression of the robustness of coherence measurement of the state in white noise environment can be obtained.The result of the robustness of coherence for quantum state at the separability boundary is consistent with the ₁l-norm coherence measurement function.