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Physica A Statistical Mechanics and its Applications

Robustness of multipartite entanglement in W and Greenberger–Horne–Zeilinger mixed states

Guo-Lin Lv, Zhu Zhen, Maoke Miao, Бо Лю, Xiaoyu Chen

Published Dec 25, 2024• 1 citation
Publisher
DOI
10.1016/j.physa.2024.130309

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The first characterization of mixed-state entanglement was achieved for two-qubit states in Werner's seminal work [Phys. Rev. A 40, 4277 (1989)]. A physically important extension concerns mixtures of a pure entangled state [such as the Greenberger-Horne-Zeilinger (GHZ) state] and the unpolarized state. These mixed states serve as benchmark for the robustness of multipartite entanglement. They share the symmetries of the GHZ state. We call such states GHZ symmetric. Here we give a complete description of the entanglement in the family of three-qubit GHZ-symmetric states and, in particular, of the three-qubit generalized Werner states. Our method relies on the appropriate parametrization of the states and on the invariance of entanglement properties under general local operations. An application is the definition of a symmetrization witness for the entanglement class of arbitrary three-qubit states.
Quantum entanglement is a fundamental characteristic of quantum mechanics, and understanding the robustness of entanglement across different mixed states is crucial for comprehending the entanglement properties of general quantum states. In this paper, the robustness of entanglement of Dicke-W and Greenberger-Horne-Zeilinger (GHZ) mixed states under different mixing ratios is calculated using the entanglement witness method. The robustnesses of entanglement of Dicke-W and GHZ mixed states are different when the probability ratio of Dicke to W is greater than 32 and less than 32. For the probability of Dicke and W states greater than or equal to 32, we study the robustness of entanglement of Dicke and GHZ mixed states and analyze and calculate their upper and lower bounds. For the probability of Dicke and W states less than 32, we take the equal probability ratio of Dicke and W states as an example and calculate and analyze the upper and lower bounds of their robustness of entanglement in detail.
We show that genuine multipartite entanglement of an arbitrary number of qubits can be witnessed sequentially for an arbitrary number of times. In the simplest scenario where only one qubit is subjected to sequential measurements, we show that genuine multiparty entanglement can be detected arbitrarily many times starting from generalized Greenberger-Horne-Zeilinger, cluster, graph, and a class of mixed states. For the case where multiple qubits are subjected to sequential measurements, we again show that detection of genuine multiparty entanglement is possible for an arbitrary number of times for multipartite Greenberger-Horne-Zeilinger and a class of mixed states.