Self-testing in the compiled setting via tilted-CHSH inequalities

TL;DR

This paper explores the limits of quantum violations of extended tilted-CHSH inequalities in a cryptographic setting, introducing a partial self-testing concept and extending sum-of-squares techniques for analysis.

Contribution

It introduces a new partial self-testing notion in the compiled setting and extends sum-of-squares methods to analyze high-degree monomials in Bell inequalities.

Findings

Quantum provers cannot significantly outperform non-communicating provers in violating these inequalities.

Extended tilted-CHSH inequalities serve as a partial self-test for the compiled setting.

The sum-of-squares technique is successfully extended to high-degree monomials in this context.

Abstract

This work investigates the family of extended tilted-CHSH inequalities in the single-prover cryptographic compiled setting. In particular, we show that a quantum polynomial-time prover can violate these Bell inequalities by at most negligibly more than the violation achieved by two non-communicating quantum provers. To obtain this result, we extend a sum-of-squares technique to monomials with arbitrarily high degree in the Bob operators and degree at most one in the Alice operators. We also introduce a notion of partial self-testing for the compiled setting, which resembles a weaker form of self-testing in the bipartite setting. As opposed to certifying the full model, partial self-testing attempts to certify the reduced states and measurements on separate subsystems. In the compiled setting, this is akin to the states after the first round of interaction and measurements made on that state. Lastly, we show that the extended tilted-CHSH inequalities satisfy this notion of a compiled self-test.