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arXiv:2104.07871 · Journal of Physics A Mathematical and Theoretical

Polytope structures for Greenberger–Horne–Zeilinger diagonal states *

Kyung Hoon Han, Seung-Hyeok Kye

Published Oct 1, 2021• 2 citations
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Abstract

Abstract We explore the polytope structures for genuine entanglement, biseparability, full biseparability and Bell inequality of multi-qubit GHZ diagonal states. We first show that biseparable GHZ diagonal states make hypersimplices inside the simplices consisting of all GHZ diagonal states. Next, we consider full biseparability which is equivalent to positive partial transpose for GHZ diagonal states, and show that they make the convex hulls of simplices and cubes. We also visualize which part of the simplex violates multipartite Bell inequality. Finally, we compute precise volumes for genuine entanglement, biseparability, full biseparability and states violating Bell inequality among all GHZ diagonal states.

DOI
10.1088/1751-8121/ac2c5a

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Abstract We explore the polytope structures for genuine entanglement, biseparability, full biseparability and Bell inequality of multi-qubit GHZ diagonal states. We first show that biseparable GHZ diagonal states make hypersimplices inside the simplices consisting of all GHZ diagonal states. Next, we consider full biseparability which is equivalent to positive partial transpose for GHZ diagonal states, and show that they make the convex hulls of simplices and cubes. We also visualize which part of the simplex violates multipartite Bell inequality. Finally, we compute precise volumes for genuine entanglement, biseparability, full biseparability and states violating Bell inequality among all GHZ diagonal states.
We analytically prove the necessary and sufficient criterion for the full separability of three-qubit Greenberger-Horne-Zeilinger (GHZ) diagonal states. The corresponding entanglement is exactly calculable for some GHZ diagonal states and is tractable for the others using the relative entropy of entanglement. We show that the biseparable criterion and the genuine entanglement are determined only by the biggest GHZ diagonal element regardless of all the other smaller diagonal elements. We have completely solved the entanglement problems of three-qubit GHZ diagonal states.
The hierarchy of nonlocality and entanglement in multipartite systems is one of the fundamental problems in quantum physics. We study this topic in three-qubit systems considering the entanglement classification of stochastic local operations and classical communication (SLOCC). The equivalence under SLOCC divides three-qubit states into separable, biseparable, WWW, and Greenberger-Horne-Zeilinger (GHZ) classes. The WWW and GHZ are two subclasses of genuine tripartite entanglement. We adopt the family of GHZ-symmetric states as a research subject, which share the symmetries of the GHZ state and have a complete characterization of SLOCC classes. In the biseparable region (with bipartite entanglement), there exist GHZ-symmetric states that are found to be fully local. In addition, there are bilocal states in both the WWW and GHZ classes. That is, neither of the subclasses of genuine tripartite entanglement can ensure genuinely tripartite nonlocality.
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.
Abstract In order to classify partial entanglement of multi-partite states, it is natural to consider the convex hulls, intersections and differences of basic convex cones obtained from partially separable states with respect to partitions of systems. In this paper, we consider convex cones consisting of -shaped three qubit states arising in this way. The class of -shaped states includes important classes like Greenberger–Horne–Zeilinger diagonal states. We find all the extreme rays of those convex cones to exhibit corresponding partially separable states. We also give characterizations for those cones which give rise to necessary criteria in terms of diagonal and anti-diagonal entries for general three qubit states.
In tri-partite systems, there are three basic biseparability, A-BC, B-CA, and C-AB, according to bipartitions of local systems. We begin with three convex sets consisting of these basic biseparable states in the three-qubit system, and consider arbitrary iterations of intersections and/or convex hulls of them to get convex cones. One natural way to classify tri-partite states is to consider those convex sets to which they belong or do not belong. This is especially useful to classify partial entanglement of mixed states. We show that the lattice generated by those three basic convex sets with respect to convex hull and intersection has infinitely many mutually distinct members to see that there are infinitely many kinds of three-qubit partial entanglement. To do this, we consider an increasing chain of convex sets in the lattice and exhibit three-qubit Greenberger–Horne–Zeilinger diagonal states distinguishing those convex sets in the chain.
A proof of Bell's theorem without inequalities valid for both inequivalent classes of three-qubit entangled states under local operations assisted by classical communication, namely Greenberger-Horne-Zeilinger (GHZ) and W,W,W, is described. This proof leads to a Bell inequality that allows more conclusive tests of Bell's theorem for three-qubit systems. Another Bell inequality involving both tri- and bipartite correlations is introduced which illustrates the different violations of local realism exhibited by the GHZ and WWW states.