Graph-Theoretic Approach for Self-Testing in Bell Scenarios

Self-testing is a technology to certify states and measurements using only the statistics of the exper-iment. Self-testing is possible if some extremal points in the set BQ of quantum correlations for a Bell experiment are achieved, up to isometries, with specific states and measurements. However, BQ is diffi-cult to characterize, so it is also difficult to prove whether or not a given matrix of quantum correlations allows for self-testing. Here, we show how some tools from graph theory can help to address this prob-lem. We observe that BQ is strictly contained in an easy-to-characterize set associated with a graph, 0(G). Therefore, whenever the optimum over BQ and the optimum over 0(G) coincide, self-testing can be demonstrated by simply proving self-testability with 0(G). Interestingly, these maxima coincide for the quantum correlations that maximally violate many families of Bell-like inequalities. Therefore, we can apply this approach to prove the self-testability of many quantum correlations, including some that are not previously known to allow for self-testing. In addition, this approach connects self-testing to some open problems in discrete mathematics. We use this connection to prove a conjecture [M. Araujo et al., Phys. Rev. A, 88, 022118 (2013)] about the closed-form expression of the Lovasz theta number for a family of graphs called the Mobius ladders. Although there are a few remaining issues (e.g., in some cases, the proof requires the assumption that measurements are of rank 1), this approach provides an alternative method to self-testing and draws interesting connections between quantum mechanics and discrete mathematics.

Automated summary

This study addresses the problem of self-testing quantum correlations using tools from graph theory. It is found that the set of quantum correlations, BQ, is strictly contained in a set associated with a graph, 0(G), which is easier to characterize. When the optimum values of BQ and 0(G) coincide, self-testing can be demonstrated by proving self-testability with 0(G). This approach also connects self-testing to open problems in discrete mathematics.