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Guess your neighbor's input: Quantum advantage in Feige's game

Simon Schmidt, Sigurd A. L. Storgaard, Michael Walter, Yuming Zhao

TL;DR

This work demonstrates a quantum advantage in Feige's two-question, three-answer nonlocal game, showing ${\\omega}_q(G_F)=9/16$ while ${\\omega}_c(G_F)=1/2$ and ${\\omega}_{ns}(G_F)=2/3$, and establishes Feige's game as a robust self-test for the 3-dimensional maximally entangled state ${|\\psi_3\\rangle}$. The authors derive tight algebraic constraints via a sums-of-squares (SOS)/NPA-style decomposition, proving a unique irreducible representation and robust self-testing behavior for the optimal quantum strategy; they also reveal an intriguing OR-structure where ${G_F=G_1\\lor G_2}$ with ${\\omega_q(G_F) > \,\\max\\{\\omega_q(G_1),\\omega_q(G_2)}}$. They further analyze parallel repetition, showing perfect non-signalling repetition for even numbers of repeats (${\\omega}_{ns}(G_F^{\\times n})=1/2^{n/2}$ for even $n$) and providing detailed classical and non-signalling bounds up to $n=4$, including a 3-fold case with known classical and non-signalling values. Finally, a synchronous variant ${G_F^s}$ is studied, revealing ${\\omega_q^s(G_F^s)=1/2}$ even though ${\\omega_q(G_F^s)=9/16}$, illustrating that quantum advantage can depend on whether strategies are required to be synchronous. Overall, the paper advances quantum-self-testing methodology, exposes a minimal entangled-state self-test, and refines understanding of quantum vs classical behavior in Feige-type nonlocal games and their parallel repetitions.

Abstract

In this article, we study a nonlocal game with two questions and three answers per player, which was first considered by Feige in 1991, and show that there is quantum advantage in this game. We prove that the game is a robust self-test for the $3$-dimensional maximally entangled state. Furthermore, we show that the game can be seen as the "or" of two games that each do not have quantum advantage. Lastly, we investigate the behavior of the game with respect to parallel repetition in the classical, quantum and non-signalling case and obtain perfect parallel repetition of the non-signalling value if Feige's game is repeated an even amount of times.

Guess your neighbor's input: Quantum advantage in Feige's game

TL;DR

This work demonstrates a quantum advantage in Feige's two-question, three-answer nonlocal game, showing while and , and establishes Feige's game as a robust self-test for the 3-dimensional maximally entangled state . The authors derive tight algebraic constraints via a sums-of-squares (SOS)/NPA-style decomposition, proving a unique irreducible representation and robust self-testing behavior for the optimal quantum strategy; they also reveal an intriguing OR-structure where with . They further analyze parallel repetition, showing perfect non-signalling repetition for even numbers of repeats ( for even ) and providing detailed classical and non-signalling bounds up to , including a 3-fold case with known classical and non-signalling values. Finally, a synchronous variant is studied, revealing even though , illustrating that quantum advantage can depend on whether strategies are required to be synchronous. Overall, the paper advances quantum-self-testing methodology, exposes a minimal entangled-state self-test, and refines understanding of quantum vs classical behavior in Feige-type nonlocal games and their parallel repetitions.

Abstract

In this article, we study a nonlocal game with two questions and three answers per player, which was first considered by Feige in 1991, and show that there is quantum advantage in this game. We prove that the game is a robust self-test for the -dimensional maximally entangled state. Furthermore, we show that the game can be seen as the "or" of two games that each do not have quantum advantage. Lastly, we investigate the behavior of the game with respect to parallel repetition in the classical, quantum and non-signalling case and obtain perfect parallel repetition of the non-signalling value if Feige's game is repeated an even amount of times.

Paper Structure

This paper contains 17 sections, 20 theorems, 214 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Strategies for nonlocal games
  4. Feige's game
  5. Construction of a quantum strategy exceeding the classical value
  6. Method for obtaining an upper bounds and algebraic relations
  7. The non-signalling value of Feige's game
  8. Feige's game is a self-test
  9. Feige's game as $(G_1\lor G_2)$-game
  10. Feige's game and parallel repetition
  11. Classical strategies for the $2m$-fold parallel repetition
  12. The non-signalling upper bound for $2m$-fold parallel repetition
  13. Values for the $3$-fold parallel repetition
  14. Feige's game as synchronous game
  15. Algebraic structure of Feige's game from sums of square decompositions
  16. ...and 2 more sections

Key Result

Theorem 1

Feige's game $G_F$ has quantum advantage with $\omega_q(G_F)=\frac{9}{16}>\frac{1}{2}=\omega_c(G_F)$.

Figures (1)

  • Figure 1: The red basis in this figure is $\{\mathinner{|\tilde{0}\rangle}, \mathinner{|\tilde{1}\rangle}, \mathinner{|\bot\rangle} \}$, the orange plane is $\mathinner{|\bot\rangle}^\perp$ and the blue plane is $\mathinner{|1\rangle}^\perp$. The green vector is $({\mathds{1}}-| \bot\rangle \langle \bot|) \mathinner{|2\rangle}$.

Theorems & Definitions (47)

  • Theorem
  • Theorem
  • Theorem
  • Theorem
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 4.1
  • ...and 37 more