Distinguishability-based genuine nonlocality with genuine multipartite entanglement

TL;DR

This paper investigates the distinguishability-based genuine nonlocality of multipartite GHZ states, revealing the existence of small genuinely nonlocal sets with linear size in local dimension and introducing the concept of threshold distinguishability.

Contribution

It introduces the concept of distinguishability-based genuine nonlocality for multipartite states and constructs small genuinely nonlocal sets of GHZ states using semidefinite programming.

Findings

Existence of small genuinely nonlocal sets of GHZ states with size linear in local dimension

Construction of (2,3)-threshold sets of GHZ states in tripartite systems

Significant gap between strong nonlocality and distinguishability-based nonlocality

Abstract

A set of orthogonal multipartite quantum states is said to be distinguishability-based genuinely nonlocal (also genuinely nonlocal, for abbreviation) if the states are locally indistinguishable across any bipartition of the subsystems. This form of multipartite nonlocality, although more naturally arising than the recently popular "strong nonlocality" in the context of local distinguishability, receives much less attention. In this work, we study the distinguishability-based genuine nonlocality of a typical type of genuine multipartite entangled states -- the d-dimensional GHZ states, featuring systems with local dimension not limited to 2. In the three-partite case, we find the existence of small genuinely nonlocal sets consisting of these states: we show that the cardinality can at least scale down to linear in the local dimension d, with the linear factor l = 1. Specifically, the method we use is semidefinite program and the GHZ states to construct these sets are special ones which we call "GHZ-lattices". This result might arguably suggest a significant gap between the strength of strong nonlocality and the distinguishability-based genuine nonlocality. Moreover, we put forward the notion of (s,n)-threshold distinguishability and utilizing a similar method, we successfully construct (2,3)-threshold sets consisting of GHZ states in three-partite systems.