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Communications in Theoretical Physics

Comparative Study of Entanglement and Wigner Function for Multi-Qubit GHZ-Squeezed State

Fatima-Zahra Siyouri

Published Dec 1, 2017• 10 citations
Publisher

Abstract

Abstract In this paper we address the possibility of using the Wigner function to capture the quantum entanglement present in a multi-qubit system. For that purpose, we calculate both the degree of entanglement and the Wigner function for mixed tripartite squeezed states of Greenberger–Horne–Zeilinger (GHZ) type then we compare their behaviors. We show that the role of Wigner function in detecting and quantifying bipartite quantum correlation [Int. J. Mod. Phys. B 30 (2016) 1650187] may be generalized to the multipartite case.

DOI
10.1088/0253-6102/68/6/729

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Abstract In this paper we address the possibility of using the Wigner function to capture the quantum entanglement present in a multi-qubit system. For that purpose, we calculate both the degree of entanglement and the Wigner function for mixed tripartite squeezed states of Greenberger–Horne–Zeilinger (GHZ) type then we compare their behaviors. We show that the role of Wigner function in detecting and quantifying bipartite quantum correlation [Int. J. Mod. Phys. B 30 (2016) 1650187] may be generalized to the multipartite case.
A comparison is made between normal discord, super quantum discord, and entanglement for the tripartite Greenberger-Horne-Zeilinger (GHZ)-type squeezed states. For this aim, we start by studying the behavior of quantum discord for tripartite GHZ-type squeezed states. Also, we verify the monotonicity of the super quantum discord according to weaker quantum measurements. This was done by manipulating the amount of quantum information lost during the disruption of our quantum system and by controlling the measurement strength. Furthermore, we calculate the entanglement for our system.
We examine the influence of photon excitation on each mode of the Glauber coherent GHZ type tripartite state. Concurrence is adopted as entanglement measure between bipartite entangled state. The pairwise concurrence is calculated and used as a quantifier of intermodal entanglement. The entanglement distribution among three modes is investigated using tangle as a measure and the residual entanglement is also calculated. The effect of the photon addition process on the quadrature squeezing is investigated. The higher order squeezing capacity of the photon addition process is also shown.
We investigate the multipartite entanglement in the phase space using the negativity of Wigner function (NWF) and in the Hilbert space using the negativity of partial transpose (NPT). We analyze comparatively these quantities and the different types of entanglements that are present in two major classes — GHZ and [Formula: see text] — made of coherent states. We show that the negativity of Wigner function can be used as a good measure of genuine entanglement in multipartite systems. However, the negativity of partial transpose is a good quantifier for only the bipartite entanglement in tripartite systems.
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 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.