Entangled Entanglement: The Geometry of GHZ States

TL;DR

This paper explores the geometric structure of GHZ states, introduces a method to generate all such states via entangled entanglement, and generalizes this approach to higher dimensions and more particles, revealing new insights into their symmetries and entanglement properties.

Contribution

It presents a novel construction of GHZ states through entangled entanglement and generalizes this to higher dimensions and multipartite systems, analyzing their geometric and entanglement features.

Findings

All 8 GHZ states form the magic simplex of entangled entanglement.

The construction reveals symmetries and cyclicity in phase operations.

The geometric analysis distinguishes regions of genuine multipartite entanglement.

Abstract

The familiar Greenberger-Horne-Zeilinger (GHZ) states can be rewritten by entangling the Bell states for two qubits with a state of the third qubit, which is dubbed entangled entanglement. We show that in this way we obtain all 8 independent GHZ states that form the simplex of entangled entanglement, the magic simplex. The construction procedure allows a generalization to higher dimensions both, in the degrees of freedom (considering qudits) as well as in the number of particles (considering n-partite states). Such bases of GHZ-type states exhibit a certain geometry that is relevant for experimental and quantum information theoretic applications. Furthermore, we study the geometry of these particular state spaces, the inherent symmetries, the cyclicity of the phase operations, and the regions of (genuine multi-partite) entanglement and the several classes of separability. We find non-trivial geometrical properties and a conceptually clear procedure to compare state spaces of different dimensions and number of particles.