Polytope structures for Greenberger-Horne-Zeilinger diagonal states

TL;DR

This paper analyzes the geometric polytope structures of multi-qubit GHZ diagonal states to understand their entanglement properties, separability, and Bell inequality violations, providing precise volume calculations.

Contribution

It introduces a geometric framework for GHZ diagonal states, characterizing various entanglement classes and Bell violations through polytope structures and volume computations.

Findings

Biseparable GHZ diagonal states form hypersimplices inside simplices.

Full biseparability corresponds to convex hulls of simplices and cubes.

Quantitative volumes for different entanglement and Bell violation regions are provided.

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.