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Entropy

Robustness of Entanglement for Dicke-W and Greenberger-Horne-Zeilinger Mixed States

Ling-Hui Zhu, Zhu Zhen, Guo-Lin Lv, Chong‐Qiang Ye, Xiaoyu Chen

Published Sep 21, 2024• 0 citations
Publisher

Abstract

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.

DOI
10.3390/e26090804

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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.
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.
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.
Entanglement detection is one of the important topics in quantum information theory. Wootters formula can be seen as a high level criterion for entanglement detection or entanglement quantification. However, it is restricted to two-qubit system, and its generalization is also limited. Based on a thorough study of a three-qubit Greenberger-Horne-Zeilinger (GHZ) state mixed with a W state and white noise, we find a connection between Wootters formula and the matched entanglement witness. Further study shows that for a four-qubit GHZ state mixed with a Dicke state and white noise, there is a new kind of Wootters formula.
Which state loses less quantum information between Greenberger-Horne-Zeilinger (GHZ) and WWW states when they are prepared for two-party quantum teleportation through a noisy channel? We address this issue by solving analytically a master equation in the Lindblad form with introducing the noisy channels that cause the quantum channels to be mixed states. It is found that the answer to this question is dependent on the type of noisy channel. If, for example, the noisy channel is (L2,x,L3,x,L4,x)({L}_{2,x},{L}_{3,x},{L}_{4,x})(L2,x​,L3,x​,L4,x​) type, where the LLL's denote the Lindblad operators, the GHZ state is always more robust than the WWW state, i.e., the GHZ state preserves more quantum information. In, however, the (L2,y,L3,y,L4,y)({L}_{2,y},{L}_{3,y},{L}_{4,y})(L2,y​,L3,y​,L4,y​)-type channel the situation becomes completely reversed. In the (L2,z,L3,z,L4,z)({L}_{2,z},{L}_{3,z},{L}_{4,z})(L2,z​,L3,z​,L4,z​)-type channel, the WWW state is more robust than the GHZ state when the noisy parameter $(\ensuremath{\kappa})$ is comparatively small while the GHZ state becomes more robust when $\ensuremath{\kappa}$ is large. In isotropic noisy channel we found that both states preserve an equal amount of quantum information. A relation between the average fidelity and entanglement for the mixed state quantum channels are discussed.
Abstract This chapter takes a closer look at entanglement in multipartite systems. An examination of pure tripartite systems serves a starting point from which we discover the equivalence classes of Greenberger-Horne-Zeilinger (GHZ) states and W states. We then continue with the mixed-state case and discuss the notions of full separability, partition-separability and biseparability versus genuine tripartite entanglement, before stating the GHZ theorem in the formulation following Mermin. For systems of three or more parties we then formulate the definitions for k-separability and genuine multipartite entanglement (GME), as well as for k-producibility and entanglement depth, followed by prominent examples for GME states such as the generalized GHZ states and Dicke states. We give an overview of the problem of detecting GME using various linear and non-linear GME witnesses, lifted witnesses, PPT mixers, before discussing challenges for the characterization and quantification of multipartite entanglement. We close by analysing the phenomenon of entangled entanglement.
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.