[Paper Review] No Strong Parallel Repetition with Entangled and Non-signaling Provers

Julia Kempe, Oded Regev|ArXiv.org|Nov 1, 2009

Complexity and Algorithms in Graphs16 references4 citations

TL;DR

This paper demonstrates that strong parallel repetition does not hold for two-prover games when provers share entanglement or are non-signaling, refuting a long-standing conjecture. It constructs explicit counterexamples using unique games and line games, showing that the winning probability decays as (1−Θ(1/n²))ℓ, proving that Holenstein's bound for non-signaling provers and known bounds for entangled provers are tight.

ABSTRACT

We consider one-round games between a classical verifier and two provers. One of the main questions in this area is the \emph{parallel repetition question}: If the game is played $\ell$ times in parallel, does the maximum winning probability decay exponentially in $\ell$? In the classical setting, this question was answered in the affirmative by Raz. More recently the question arose whether the decay is of the form $(1-Θ(\eps))^\ell$ where $1-\eps$ is the value of the game and $\ell$ is the number of repetitions. This question is known as the \emph{strong parallel repetition question} and was motivated by its connections to the unique games conjecture. It was resolved by Raz who showed that strong parallel repetition does \emph{not} hold, even in the very special case of games known as XOR games. This opens the question whether strong parallel repetition holds in the case when the provers share entanglement. Evidence for this is provided by the behavior of XOR games, which have strong (in fact \emph{perfect}) parallel repetition, and by the recently proved strong parallel repetition of linear unique games. A similar question was open for games with so-called non-signaling provers. Here the best known parallel repetition theorem is due to Holenstein, and is of the form $(1-Θ(\eps^2))^\ell$. We show that strong parallel repetition holds neither with entangled provers nor with non-signaling provers. In particular we obtain that Holenstein's bound is tight. Along the way we also provide a tight characterization of the asymptotic behavior of the entangled value under parallel repetition of unique games in terms of a semidefinite program.

Motivation & Objective

• To resolve the strong parallel repetition question for games with entangled and non-signaling provers.
• To determine whether the winning probability in repeated games decays exponentially as (1−ε)ℓ for games with entangled or non-signaling provers.
• To establish tight bounds on the decay rate of the entangled and non-signaling values under parallel repetition.
• To provide a characterization of the asymptotic behavior of the entangled value of unique games via semidefinite programming.
• To show that existing parallel repetition theorems for non-signaling and entangled provers are optimal by constructing tight counterexamples.

Proposed method

• Constructs a family of unique games, the 'unique line game' G_uL, with classical value ω(G_uL) = 1−Θ(1/n), entangled value ω*(G_uL) = 1−Θ(1/n²), and non-signaling value ω^ns(G_uL) = 1−Θ(1/n²).
• Uses semidefinite programming (SDP1 and SDP2) to characterize the entangled value of unique games, showing a quadratic gap between SDP1 and SDP2 solutions.
• Applies a tensor product strategy based on correlated sampling and orthonormal bases derived from SDP vectors to construct a winning strategy for repeated games.
• Demonstrates that the strategy for G_uL^ℓ is valid for the line game G_L^ℓ by showing it never uses answer 3 and preserves winning probability.
• Leverages the fact that the SDP solution for G_uL has non-negative inner products to ensure feasibility under SDP2 constraints.
• Proves tightness of Holenstein's bound by showing that ω^ns(G_L^ℓ) ≥ (1−O(1/n²))ℓ for ℓ ≥ n², matching the decay rate of the non-signaling value.

Experimental results

Research questions

• RQ1 Does strong parallel repetition hold for two-prover games when provers share entanglement?
• RQ2 Is the decay rate of the non-signaling value under parallel repetition of the form (1−Θ(ε))ℓ, or is it slower?
• RQ3 Can the parallel repetition bound of Holenstein for non-signaling provers be improved, or is it tight?
• RQ4 What is the asymptotic behavior of the entangled value of unique games under parallel repetition?
• RQ5 Do XOR games or other special classes of games exhibit perfect or strong parallel repetition in the entangled or non-signaling setting?

Key findings

• Strong parallel repetition does not hold for entangled provers, as shown by constructing a family of unique games where the entangled value decays as (1−Θ(1/n²))ℓ.
• The non-signaling value of the same games also decays as (1−Θ(1/n²))ℓ, proving that Holenstein's bound of (1−Θ(ε²))ℓ is tight.
• The line game G_L provides a counterexample to strong parallel repetition with alphabet size 2, showing that even simple games can violate strong parallel repetition.
• A quadratic gap exists between the SDP1 and SDP2 relaxations for unique games, demonstrating that SDP2 is strictly stronger than SDP1 in this context.
• The constructed strategy for repeated games is not a product strategy due to a correlated sampling step, despite being based on a tensor product of SDP solutions.
• The classical, entangled, and non-signaling values of G_L^ℓ all satisfy ω(G_L^ℓ) ≥ (1−O(1/n²))ℓ for ℓ ≥ n², showing that the decay rate is inherent to the game structure.