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

TL;DR

This work addresses certifying quantum correlations in a single-prover compiled setting produced by a quantum homomorphic encryption (QHE) compilation, relating compiled strategies to bipartite quantum bounds. The authors extend the sum-of-squares (SOS) framework to allow arbitrary Bob monomials and degree-one Alice components, realized through extended pseudo-expectations, to upper-bound the compiled value $ I$ by the bipartite quantum bound $^Q$ up to a negligible function of the security parameter. They also define compiled self-testing based on partial models and compiled-counterparts, proving that the extended tilted-CHSH family with parameters $( heta,\,\

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.

Theorems & Definitions (49)

• Lemma 3.1: kulpe24, Proposition 4.6
• Lemma 3.1: kulpe24, Proposition 4.6
• Lemma 3.2
• Theorem 3.3
• Lemma 3.4: Barizien2024, Section 3.2.1
• Theorem 3.5
• proof
• Remark 3.6
• Definition 4.1: Bipartite self-test
• Definition 4.2: Partial model