Einstein-Podolsky-Rosen correlations and Bell correlations in the simplest scenario

TL;DR

This paper explores the relationship between EPR steering and Bell nonlocality in the simplest two-qubit measurement scenario, revealing conditions under which states exhibit these quantum correlations and introducing criteria for their detection.

Contribution

It establishes that in the simplest scenario, EPR-nonlocal full correlations coincide with Bell-nonlocal correlations, and introduces a semidefinite-programming criterion for steerability detection.

Findings

EPR-nonlocal correlations occur if and only if Bell-nonlocal correlations occur in this scenario.

The same scenario can reveal one-way steering and hierarchy between steering and Bell nonlocality.

A necessary and sufficient semidefinite-programming criterion for steerability is derived.

Abstract

Einstein-Podolsky-Rosen (EPR) steering is an intermediate type of quantum nonlocality which sits between entanglement and Bell nonlocality. A set of correlations is Bell nonlocal if it does not admit a local hidden variable (LHV) model, while it is EPR nonlocal if it does not admit a local hidden variable-local hidden state (LHV-LHS) model. It is interesting to know what states can generate EPR-nonlocal correlations in the simplest nontrivial scenario, that is, two projective measurements for each party sharing a two-qubit state. Here we show that a two-qubit state can generate EPR-nonlocal full correlations (excluding marginal statistics) in this scenario if and only if it can generate Bell-nonlocal correlations. If full statistics (including marginal statistics) is taken into account, surprisingly, the same scenario can manifest the simplest one-way steering and the strongest hierarchy between steering and Bell nonlocality. To illustrate these intriguing phenomena in simple setups, several concrete examples are discussed in detail, which facilitates experimental demonstration. In the course of study, we introduce the concept of restricted LHS models and thereby derive a necessary and sufficient semidefinite-programming criterion to determine the steerability of any bipartite state under given measurements. Analytical criteria are further derived in several scenarios of strong theoretical and experimental interest.