Non-local games and optimal steering at the boundary of the quantum set

TL;DR

This paper investigates the boundary of the quantum set of correlations, demonstrating that optimal steering properties extend to many boundary points and clarifying the conditions under which these properties hold.

Contribution

The authors prove that the optimal steering property identified by Oppenheim and Wehner applies to a broad class of quantum boundary points, including some previously thought to be exceptions.

Findings

The optimal steering property holds for many points on the quantum boundary.

The property does not always respect no-signaling equivalence, requiring specific Bell expression forms.

The results extend understanding of quantum correlations and steering at the boundary.

Abstract

The boundary between classical and quantum correlations is well characterised by linear constraints called Bell inequalities. It is much harder to characterise the boundary of the quantum set itself in the space of no-signaling correlations. For the points on the quantum boundary that violate maximally some Bell inequalities, Oppenheim and Wehner [Science 330, 1072 (2010)] pointed out a complex property: the optimal measurements of Alice steer Bob's local state to the eigenstate of an effective operator corresponding to its maximal eigenvalue. This effective operator is the linear combination of Bob's local operators induced by the coefficients of the Bell inequality, and it can be interpreted as defining a fine-grained uncertainty relation. It is natural to ask whether the same property holds for other points on the quantum boundary, using the Bell expression that defines the tangent hyperplane at each point. We prove that this is indeed the case for a large set of points, including some that were believed to provide counterexamples. The price to pay is to acknowledge that the Oppenheim-Wehner criterion does not respect equivalence under the no-signaling constraint: for each point, one has to look for specific forms of writing the Bell expressions.