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A Noncommutative Nullstellensatz for Perfect Two-Answer Quantum Nonlocal Games

Tianshi Yu, Lihong Zhi

TL;DR

This work introduces a noncommutative Nullstellensatz in the context of perfect two-answer quantum nonlocal games. It proves that a two-answer game with a perfect commuting-operator strategy also has a perfect classical strategy, extending finite-dimensional results to infinite-dimensional settings. The result is formulated via the universal game algebra $\mathcal{A}$, a left ideal $\mathcal{L}(\mathcal{N})$, and an Archimedean sums-of-squares cone $\mathrm{SOS}_{\mathcal{A}}$, yielding a one-dimensional representation that annihilates the invalid set $\mathcal{N}$; this establishes a noncommutative Nullstellensatz in this special case and links commuting-operator strategies to deterministic classical strategies. The findings illuminate the algebraic structure of perfect strategies and delineate the scope and limits of the approach, especially noting the constraint to two answers and infinite-dimensional settings.

Abstract

This paper introduces a noncommutative version of the Nullstellensatz, motivated by the study of quantum nonlocal games. It has been proved that a two-answer nonlocal game with a perfect quantum strategy also admits a perfect classical strategy. We generalize this result to the infinite-dimensional case, showing that a two-answer game with a perfect commuting operator strategy also admits a perfect classical strategy. This result induces a special case of noncommutative Nullstellensatz.

A Noncommutative Nullstellensatz for Perfect Two-Answer Quantum Nonlocal Games

TL;DR

This work introduces a noncommutative Nullstellensatz in the context of perfect two-answer quantum nonlocal games. It proves that a two-answer game with a perfect commuting-operator strategy also has a perfect classical strategy, extending finite-dimensional results to infinite-dimensional settings. The result is formulated via the universal game algebra , a left ideal , and an Archimedean sums-of-squares cone , yielding a one-dimensional representation that annihilates the invalid set ; this establishes a noncommutative Nullstellensatz in this special case and links commuting-operator strategies to deterministic classical strategies. The findings illuminate the algebraic structure of perfect strategies and delineate the scope and limits of the approach, especially noting the constraint to two answers and infinite-dimensional settings.

Abstract

This paper introduces a noncommutative version of the Nullstellensatz, motivated by the study of quantum nonlocal games. It has been proved that a two-answer nonlocal game with a perfect quantum strategy also admits a perfect classical strategy. We generalize this result to the infinite-dimensional case, showing that a two-answer game with a perfect commuting operator strategy also admits a perfect classical strategy. This result induces a special case of noncommutative Nullstellensatz.
Paper Structure (6 sections, 3 theorems, 83 equations)

This paper contains 6 sections, 3 theorems, 83 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Motivations
  4. Universal Game Algebra for Two-Answer Games
  5. Main Results
  6. Some Discussions

Key Result

Theorem 3.1

Let $\mathcal{A}$ denote the universal game algebra for two-answer games. Let be the index set of $\mathcal{N}$, where Let $\mathcal{L}(\mathcal{N})$ be the left ideal generated by $\mathcal{N}$. Then if and only if there exists a $*-$representation such that

Theorems & Definitions (15)

  • Definition 2.1
  • Theorem 3.1
  • Remark 1
  • Proposition 3.2
  • proof : Proof Sketch
  • Proposition 3.3
  • Remark 2
  • proof
  • Claim 3.4
  • Remark 3
  • ...and 5 more