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Quantum Information Processing

Robustness of entanglement for W and Greenberger–Horne–Zeilinger mixed states

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

Published Dec 26, 2024• 1 citation
Publisher

Abstract

Abstract We study the robustness of entanglement for W and GHZ mixed states of three and four qubits. By using the definition of the robustness of entanglement and constructing multi-partite fully separable states, we obtain the analytical upper bound of the robustness of entanglement. By analyzing the entanglement witness , we obtain the lower bound of the robustness of entanglement. Then we find that the upper bound and the lower bound are fully coincident, which shows that our results are accurate and valuable.

DOI
10.1007/s11128-024-04620-6

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Abstract We study the robustness of entanglement for W and GHZ mixed states of three and four qubits. By using the definition of the robustness of entanglement and constructing multi-partite fully separable states, we obtain the analytical upper bound of the robustness of entanglement. By analyzing the entanglement witness , we obtain the lower bound of the robustness of entanglement. Then we find that the upper bound and the lower bound are fully coincident, which shows that our results are accurate and valuable.
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.
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.
Abstract We characterize the separability of three qubit GHZ diagonal states in terms of entries. This enables us to check separability of GHZ diagonal states without decomposition into the sum of pure product states. In the course of discussion, we show that the necessary criterion of Gühne (2011 Entanglement criteria and full separability of multi-qubit quantum states Phys. Lett. A 375 406–10 ) for (full) separability of three qubit GHZ diagonal states is sufficient with a simpler formula. The main tool is to use entanglement witnesses which are tri-partite Choi matrices of positive bi-linear maps.
Recently, a new type of symmetry for three-qubit quantum states was introduced, the so-called Greenberger-Horne-Zeilinger (GHZ) symmetry. It includes the operations which leave the three-qubit standard GHZ state unchanged. This symmetry is powerful as it yields families of mixed states that are, on the one hand, complex enough from the physics point of view and, on the other hand, simple enough mathematically so that their properties can be characterized analytically. We show that by using the properties of GHZ-symmetric states it is straightforward to derive optimal witnesses for three-qubit entanglement.
We study the robustness of multiparty entanglement under local decoherence, modeled by partially depolarizing channels acting independently on each subsystem. Surprisingly, we find that the robustness of (GHZ) entanglement increases with the number of qubits. An n-qubit Greenberger-Horne-Zeilinger (GHZ) state can stand more than 55% local depolarization in the limit $\stackrel{\ensuremath{\rightarrow}}{n}\ensuremath{\infty}.$ GHZ states are more robust than other generic states of three and four qubits. We also study spin-squeezed states in the limit $\stackrel{\ensuremath{\rightarrow}}{n}\ensuremath{\infty}$ and find that they too can stand considerable local depolarization.
In this article the robustness of tripartite Greenberger-Horne-Zeilinger (GHZ) and W states is investigated against Dzyaloshinskii-Moriya (i.e. DM) interaction. We consider a closed system of three qubits and an environmental qubit. The environmental qubit interacts with any one of the three qubits through DM interaction. The tripartite system is initially prepared in GHZ and W states respectively. The composite four qubits system evolve with unitary dynamics. We detach the environmental qubit by tracing out from four qubits and profound impact of DM interaction is studied on the initial entanglement of the system. As a result we find that the bipartite partitions of W states suffer from entanglement sudden death (i.e. ESD), while tripartite entanglement does not. On the other hand bipartite partitions and tripartite entanglement in GHZ states does not feel any influence of DM interaction. So, we find that GHZ states have robust character than W states. In this work we consider generalised GHZ and W states and three πππ is used as an entanglement measure. This study can be useful in quantum information processing where unwanted DM interaction takes place.