A geometric comparison of entanglement and quantum nonlocality in discrete systems

TL;DR

This paper explores the geometric relationship between entanglement and quantum nonlocality in bipartite qudits, revealing their differences through geometric illustrations and numerical analysis.

Contribution

It introduces a geometric framework for comparing entanglement and nonlocality, highlighting their non-monotonic relationship and differences in state space boundaries.

Findings

Boundaries of separability and Bell violation are geometrically distinct.

Bell violations and entanglement are non-monotonically related for mixed states.

Geometric illustrations emphasize the difference between entanglement and nonlocality.

Abstract

We compare entanglement with quantum nonlocality employing a geometric structure of the state space of bipartite qudits. Central object is a regular simplex spanned by generalized Bell states. The Collins-Gisin-Linden-Massar-Popescu-Bell inequality is used to reveal states of this set that cannot be described by local-realistic theories. Optimal measurement settings necessary to ascertain nonlocality are determined by means of a recently proposed parameterization of the unitary group U(d) combined with robust numerical methods. The main results of this paper are descriptive geometric illustrations of the state space that emphasize the difference between entanglement and quantum nonlocality. Namely, it is found that the shape of the boundaries of separability and Bell inequality violation are essentially different. Moreover, it is shown that also for mixtures of states sharing the same amount of entanglement, Bell inequality violations and entanglement measures are non-monotonically related.