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Robust Self-testing for Synchronous Correlations and Games

Prem Nigam Kar

TL;DR

The paper presents an operator-algebraic framework for robust self-testing of synchronous correlations and games, showing that robustness is equivalent to the existence of a unique implementing state on the minimised tensor-product $C^*$-algebra $\mathcal{A}_{XA}^{POVM} \tens_{\min} \mathcal{A}_{XA}^{POVM}$. It extends this characterization to synchronous games via unique amenable tracial states on the corresponding $C^*$-algebra, and proves that all robust self-tests arise from such unique states, including a first explicit separation between robust self-testing and commuting-operator self-testing. The work then applies these ideas to Linear Constraint System games and demonstrates the existence of a synchronous robust self-test that is not a commuting-operator self-test, thereby clarifying the landscape of self-testing across operator models. Overall, the results connect robust self-testing to abstract state self-testing, residually finite-dimensional algebras, and amenable traces, offering a principled route to proving robust self-testing in a broad class of synchronous correlations and games with potential implications for device-independent protocols and quantum complexity.

Abstract

We develop an abstract operator-algebraic characterization of robust self-testing for synchronous correlations and games. Specifically, we show that a synchronous correlation is a robust self-test if and only if there is a unique state on an appropriate $C^*$-algebra that "implements" the correlation. Extending this result, we prove that a synchronous game is a robust self-test if and only if its associated $C^*$-algebra admits a unique amenable tracial state. This framework allows us to establish that all synchronous correlations and games that serve as commuting operator self-tests for finite-dimensional strategies are also robust self-tests. As an application, we recover sufficient conditions for linear constraint system games to exhibit robust self-testing. We also demonstrate the existence of a synchronous nonlocal game that is a robust self-test but not a commuting operator self-test, showing that these notions are not equivalent.

Robust Self-testing for Synchronous Correlations and Games

TL;DR

The paper presents an operator-algebraic framework for robust self-testing of synchronous correlations and games, showing that robustness is equivalent to the existence of a unique implementing state on the minimised tensor-product -algebra . It extends this characterization to synchronous games via unique amenable tracial states on the corresponding -algebra, and proves that all robust self-tests arise from such unique states, including a first explicit separation between robust self-testing and commuting-operator self-testing. The work then applies these ideas to Linear Constraint System games and demonstrates the existence of a synchronous robust self-test that is not a commuting-operator self-test, thereby clarifying the landscape of self-testing across operator models. Overall, the results connect robust self-testing to abstract state self-testing, residually finite-dimensional algebras, and amenable traces, offering a principled route to proving robust self-testing in a broad class of synchronous correlations and games with potential implications for device-independent protocols and quantum complexity.

Abstract

We develop an abstract operator-algebraic characterization of robust self-testing for synchronous correlations and games. Specifically, we show that a synchronous correlation is a robust self-test if and only if there is a unique state on an appropriate -algebra that "implements" the correlation. Extending this result, we prove that a synchronous game is a robust self-test if and only if its associated -algebra admits a unique amenable tracial state. This framework allows us to establish that all synchronous correlations and games that serve as commuting operator self-tests for finite-dimensional strategies are also robust self-tests. As an application, we recover sufficient conditions for linear constraint system games to exhibit robust self-testing. We also demonstrate the existence of a synchronous nonlocal game that is a robust self-test but not a commuting operator self-test, showing that these notions are not equivalent.

Paper Structure

This paper contains 14 sections, 30 theorems, 52 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Nonlocal Games
  4. Self-testing
  5. Background on Operator Algebras
  6. Abstract State Self-Testing
  7. Characterising Synchronous Self-tests
  8. Characterising Robust Synchronous Self-Tests
  9. Analogue of Gowers-Hatami
  10. Necessary and Sufficient Conditions for Robust Self-testing
  11. Game Self-tests
  12. Nonlocal Games that Robustly Self-Test a Strategy
  13. Separating Robust Self-tests and Commuting Operator Self-Tests
  14. Discussion

Key Result

Theorem 1.5

Let $\widetilde{p} = \{\widetilde{p}(a,b\mid x,y)\}_{x,y \in X, a, b \in A}$ be a synchronous quantum correlation that is an extreme point of $C_q(X,X,A,A)$. Then, the following are equivalent:

Theorems & Definitions (48)

  • Theorem 1.5: Main Result
  • Proposition 1.6
  • Theorem 1.7
  • Definition 2.1
  • Proposition 2.2: helton2017algebras
  • Lemma 3.1: paddock_operator-algebraic_2023
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • ...and 38 more