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Quantum bounds for compiled XOR games and $d$-outcome CHSH games

Matilde Baroni, Quoc-Huy Vu, Boris Bourdoncle, Eleni Diamanti, Damian Markham, Ivan Šupić

TL;DR

The paper investigates whether Kalai et al.'s quantum-homomorphic-encryption-based compiler preserves the quantum bound for nonlocal games beyond CHSH in a single-prover setting. Using a cryptographic sum-of-squares (SOS) decomposition and a pseudo-expectation map, the authors transfer nonlocal bounds to compiled games while controlling security via a negligible function $\delta_{\mathrm{QHE}}(\kappa)$, enabling rigorous upper bounds on compiled scores. They prove that the quantum bound is preserved for XOR games and for $d$-outcome SATWAP/CHSH-type inequalities, and they establish computational self-testing results for pairs of qubit measurements and for three Pauli observables within compiled protocols. The framework combines SOS techniques, NPA-type reasoning, and cryptographic guarantees to enable device-independent certification on a single quantum device, with implications for practical quantum certification and cryptographic protocols; concurrent work and several open directions are also discussed.

Abstract

Nonlocal games play a crucial role in quantum information theory and have numerous applications in certification and cryptographic protocols. Kalai et al. (STOC 2023) introduced a procedure to compile a nonlocal game into a single-prover interactive proof, using a quantum homomorphic encryption scheme, and showed that their compilation method preserves the classical bound of the game. Natarajan and Zhang (FOCS 2023) then showed that the quantum bound is preserved for the specific case of the CHSH game. Extending the proof techniques of Natarajan and Zhang, we show that the compilation procedure of Kalai et al. preserves the quantum bound for two classes of games: XOR games and d-outcome CHSH games. We also establish that, for any pair of qubit measurements, there exists an XOR game such that its optimal winning probability serves as a self-test for that particular pair of measurements.

Quantum bounds for compiled XOR games and $d$-outcome CHSH games

TL;DR

The paper investigates whether Kalai et al.'s quantum-homomorphic-encryption-based compiler preserves the quantum bound for nonlocal games beyond CHSH in a single-prover setting. Using a cryptographic sum-of-squares (SOS) decomposition and a pseudo-expectation map, the authors transfer nonlocal bounds to compiled games while controlling security via a negligible function , enabling rigorous upper bounds on compiled scores. They prove that the quantum bound is preserved for XOR games and for -outcome SATWAP/CHSH-type inequalities, and they establish computational self-testing results for pairs of qubit measurements and for three Pauli observables within compiled protocols. The framework combines SOS techniques, NPA-type reasoning, and cryptographic guarantees to enable device-independent certification on a single quantum device, with implications for practical quantum certification and cryptographic protocols; concurrent work and several open directions are also discussed.

Abstract

Nonlocal games play a crucial role in quantum information theory and have numerous applications in certification and cryptographic protocols. Kalai et al. (STOC 2023) introduced a procedure to compile a nonlocal game into a single-prover interactive proof, using a quantum homomorphic encryption scheme, and showed that their compilation method preserves the classical bound of the game. Natarajan and Zhang (FOCS 2023) then showed that the quantum bound is preserved for the specific case of the CHSH game. Extending the proof techniques of Natarajan and Zhang, we show that the compilation procedure of Kalai et al. preserves the quantum bound for two classes of games: XOR games and d-outcome CHSH games. We also establish that, for any pair of qubit measurements, there exists an XOR game such that its optimal winning probability serves as a self-test for that particular pair of measurements.
Paper Structure (20 sections, 17 theorems, 127 equations, 1 figure)

This paper contains 20 sections, 17 theorems, 127 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Our Contributions
  3. Outline of the paper
  4. Concurrent and independent work
  5. Subsequent works
  6. Preliminaries
  7. Nonlocal games and Bell inequalities
  8. XOR games
  9. Self-testing
  10. Quantum homomorphic encryption
  11. Compiled nonlocal games
  12. Modelling the quantum prover
  13. Technical tools for estimating quantum bounds of compiled nonlocal games
  14. Block encodings
  15. Crypto-correlation matrix and pseudo-expectation map
  16. ...and 5 more sections

Key Result

Lemma 1

Alice's optimal quantum strategy in an XOR game, encoded in the vector ${\lvert A_q\rangle}$, is fully determined by Bob's optimal strategy ${\lvert B_q\rangle}$ through a linear transformation:

Figures (1)

  • Figure 1: Pictorial representation of the Kalai et. al. compilation protocol for $2$-player nonlocal games. On the left, a general $2$-player nonlocal game; the two parties are spatially separated, and only communicate to a classical verifier which is sampling questions $(x,y)$ and collecting their answers $(a,b)$. On the right, the single prover game resulting from the compilation procedure. In this representation, time flows downwards.

Theorems & Definitions (33)

  • Lemma 1: tsirelson
  • proof
  • Theorem 1
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • proof
  • Definition 1: Quantum polynomial time algorithm
  • Definition 2: Quantum Homomorphic Encryption (QHE)
  • ...and 23 more