Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A bound on the quantum value of all compiled nonlocal games

Alexander Kulpe, Giulio Malavolta, Connor Paddock, Simon Schmidt, Michael Walter

TL;DR

This work establishes a general quantum-soundness bound for compiled two-player nonlocal games: the asymptotic quantum value of the compiled game cannot exceed the original game's quantum commuting value $\omega_{qc}(\mathcal{G})$. The authors fuse operator-algebra techniques with cryptographic security notions, showing that computational non-signaling in the compiled protocol yields a limiting commuting-operator correlation via a universal $C^*$-algebra and GNS framework. They also prove an asymptotic self-testing statement for compiled games when the base game is a commuting-operator self-test (e.g., CHSH). The results resolve a long-standing open problem by giving a universal bound for all compiled two-player games and outline avenues for tighter bounds, multi-prover generalizations, and potential weaker cryptographic assumptions.

Abstract

A cryptographic compiler introduced by Kalai et al. (STOC'23) converts any nonlocal game into an interactive protocol with a single computationally bounded prover. Although the compiler is known to be sound in the case of classical provers and complete in the quantum case, quantum soundness has so far only been established for special classes of games. In this work, we establish a quantum soundness result for all compiled two-player nonlocal games. In particular, we prove that the quantum commuting operator value of the underlying nonlocal game is an upper bound on the quantum value of the compiled game. Our result employs techniques from operator algebras in a computational and cryptographic setting to establish information-theoretic objects in the asymptotic limit of the security parameter. It further relies on a sequential characterization of quantum commuting operator correlations which may be of independent interest.

A bound on the quantum value of all compiled nonlocal games

TL;DR

This work establishes a general quantum-soundness bound for compiled two-player nonlocal games: the asymptotic quantum value of the compiled game cannot exceed the original game's quantum commuting value . The authors fuse operator-algebra techniques with cryptographic security notions, showing that computational non-signaling in the compiled protocol yields a limiting commuting-operator correlation via a universal -algebra and GNS framework. They also prove an asymptotic self-testing statement for compiled games when the base game is a commuting-operator self-test (e.g., CHSH). The results resolve a long-standing open problem by giving a universal bound for all compiled two-player games and outline avenues for tighter bounds, multi-prover generalizations, and potential weaker cryptographic assumptions.

Abstract

A cryptographic compiler introduced by Kalai et al. (STOC'23) converts any nonlocal game into an interactive protocol with a single computationally bounded prover. Although the compiler is known to be sound in the case of classical provers and complete in the quantum case, quantum soundness has so far only been established for special classes of games. In this work, we establish a quantum soundness result for all compiled two-player nonlocal games. In particular, we prove that the quantum commuting operator value of the underlying nonlocal game is an upper bound on the quantum value of the compiled game. Our result employs techniques from operator algebras in a computational and cryptographic setting to establish information-theoretic objects in the asymptotic limit of the security parameter. It further relies on a sequential characterization of quantum commuting operator correlations which may be of independent interest.
Paper Structure (26 sections, 18 theorems, 70 equations, 1 figure)

This paper contains 26 sections, 18 theorems, 70 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Background on compiled nonlocal games
  3. Main results
  4. Technical outline
  5. Open problems and outlook
  6. Organization of the paper
  7. Preliminaries
  8. Vectors, operators, quantum mechanics
  9. Algebras and representations
  10. Classical and quantum computing
  11. Nonlocal games and strategies
  12. Nonlocal games
  13. Strategies and correlations
  14. Values of games
  15. Compiled nonlocal games
  16. ...and 11 more sections

Key Result

Theorem 1.1

For large enough security parameter $\lambda$, no QPT strategy can win the compiled game with probability exceeding the quantum commuting operator value of the game by any constant.

Figures (1)

  • Figure 1: \ref{['sfig:nonlocal_game']} depicts a conventional nonlocal game between a verifier $\mathsf{V}$ and non-communicating computationally unbounded quantum provers $\mathsf{A}$ and $\mathsf{B}$. \ref{['sfig:comp_game']} depicts a compiled game with verifier $\mathsf{V}$ and a single computationally bounded quantum prover $\mathsf{P}$. The security of the encryption scheme ensures that post measurement states following the first round of communication are computationally indistinguishable. \ref{['sfig:seq_game']} depicts a sequential game, which is an idealized compiled game, with a verifier $\mathsf{V}$ and a single quantum prover $\mathsf{P}$. In a sequential game we assume that all post-measurement states after the first round of communication are identical.

Theorems & Definitions (48)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 3.1
  • Remark 3.2
  • Definition 3.3
  • Definition 3.4
  • Definition 3.5
  • Definition 3.6
  • Definition 3.7
  • Definition 4.1
  • ...and 38 more