Steering Bell-diagonal states

TL;DR

This paper completely characterizes the steerability of Bell-diagonal states under one and two projective measurements, revealing a direct link to CHSH inequality violation and exploring connections with entanglement and Bell nonlocality.

Contribution

It provides a necessary and sufficient criterion for steerability of Bell-diagonal states with two projective measurements and introduces a steering measure related to concurrence and ellipsoid volume.

Findings

Bell-diagonal states are steerable iff they violate CHSH inequality.

Maximal concurrence and ellipsoid volume for unsteerable Bell-diagonal states are determined.

A sufficient criterion for steerability under three projective measurements is proposed.

Abstract

We investigate the steerability of two-qubit Bell-diagonal states under projective measurements by the steering party. In the simplest nontrivial scenario of two projective measurements, we solve this problem completely by virtue of the connection between the steering problem and the joint-measurement problem. A necessary and sufficient criterion is derived together with a simple geometrical interpretation. Our study shows that a Bell-diagonal state is steerable by two projective measurements iff it violates the Clauser-Horne-Shimony-Holt (CHSH) inequality, in sharp contrast with the strict hierarchy expected between steering and Bell nonlocality. We also introduce a steering measure and clarify its connections with concurrence and the volume of the steering ellipsoid. In particular, we determine the maximal concurrence and ellipsoid volume of Bell-diagonal states that are not steerable by two projective measurements. Finally, we explore the steerability of Bell-diagonal states under three projective measurements. A simple sufficient criterion is derived, which can detect the steerability of many states that are not steerable by two projective measurements. Our study offers valuable insight on steering of Bell-diagonal states as well as the connections between entanglement, steering, and Bell nonlocality.