Separation of quantum, spatial quantum, and approximate quantum correlations

Salman Beigi

doi:10.22331/q-2021-01-28-389

Abstract

Quantum nonlocal correlations are generated by implementation of local quantum measurements on spatially separated quantum subsystems. Depending on the underlying mathematical model, various notions of sets of quantum correlations can be defined. In this paper we prove separations of such sets of quantum correlations. In particular, we show that the set of bipartite quantum correlations with four binary measurements per party becomes strictly smaller once we restrict the local Hilbert spaces to be finite dimensional, i.e., Cq(4,4,2,2)≠Cqs(4,4,2,2). We also prove non-closure of the set of bipartite quantum correlations with four ternary measurements per party, i.e., Cqs(4,4,3,3)≠Cqa(4,4,3,3).

Highlights

Nonlocality is one of the most fascinating features of quantum physics, stating that spatially separated parties can generate correlations that cannot be generated in the local hidden variable model [10]
In a bipartite nonlocality scenario à la Bell [3], it is assumed that two parties, Alice and Bob can apply a measurement of their choice, which we denote by labels s and t respectively, on their respective subsystems
Sarkar et al in this paper show that any maximally entangled state of Schmidt rank d can be self-tested via the Bell inequality of [16]

Summary

Introduction

Nonlocality is one of the most fascinating features of quantum physics, stating that spatially separated parties can generate correlations that cannot be generated in the local hidden variable model [10]. Separation of Cq and Cqs was first conjectured in [15] and proved by Coladangelo and Stark in [7] Their proof is quite elementary and is based on the idea of self-testing of the so called tilted CHSH inequality [1]. After this modification, the argument of [7] based on the analysis of Schmidt coefficients of the shared state does not directly work To overcome this difficulty, we further use self-testing properties of tilted CHSH correlations and build our arguments based on the ranks of the measurement operators. We further use self-testing properties of tilted CHSH correlations and build our arguments based on the ranks of the measurement operators This gives us our first result Cq(4,4,2,2) = Cq(4s,4,2,2). In the following two sections we will prove Theorem 1 and Theorem 2

Self-testing protocols

Quantum correlations in finite vs infinite dimensions

Non-closure of the set of quantum correlations