A parallel repetition theorem for entangled projection games

We study the behavior of the entangled value of two-player one-round projection games under parallel repetition. We show that for any projection game $G$ of entangled value 1-eps < 1, the value of the $k$-fold repetition of G goes to zero as O((1-eps^c)^k), for some universal constant c\geq 1. Previously parallel repetition with an exponential decay in $k$ was only known for the case of XOR and unique games. To prove the theorem we extend an analytical framework recently introduced by Dinur and Steurer for the study of the classical value of projection games under parallel repetition. Our proof, as theirs, relies on the introduction of a simple relaxation of the entangled value that is perfectly multiplicative. The main technical component of the proof consists in showing that the relaxed value remains tightly connected to the entangled value, thereby establishing the parallel repetition theorem. More generally, we obtain results on the behavior of the entangled value under products of arbitrary (not necessarily identical) projection games. Relating our relaxed value to the entangled value is done by giving an algorithm for converting a relaxed variant of quantum strategies that we call “vector quantum strategy” to a quantum strategy. The algorithm is considerably simpler in case the bipartite distribution of questions in the game has good expansion properties. When this is not the case, rounding relies on a quantum analogue of Holenstein’s correlated sampling lemma which may be of independent interest. Our “quantum correlated sampling lemma” generalizes results of van Dam and Hayden on universal embezzlement.

💡 Research Summary

The paper establishes a general parallel‑repetition theorem for two‑player one‑round projection games when the players are allowed to share quantum entanglement. For any projection game G whose entangled value VAL*(G) is strictly less than 1, the value of the k‑fold parallel repetition G^{⊗k} decays exponentially as O((1‑ε^{c})^{k}) for a universal constant c≥1 (and C>0), and when the underlying question distribution has good expansion the optimal exponent c=1 is achieved.

The authors achieve this by extending the analytical framework of Dinur and Steurer, originally designed for the classical value, to the quantum setting. They introduce a relaxed quantity VAL*⁺(G), defined via “vector quantum strategies”. A vector quantum strategy assigns a complex vector to each question‑answer pair; the game operator is applied to these vectors and the resulting inner products are maximized. Crucially, VAL*⁺ is perfectly multiplicative: VAL*⁺(G⊗H)=VAL*⁺(G)·VAL*⁺(H).

The second major ingredient is a rounding procedure that converts a high‑valued vector quantum strategy into an actual quantum strategy (i.e., a shared entangled state together with local measurements) without losing more than a constant factor. When the bipartite question distribution is an expander, simple normalization suffices. In the general case the authors develop a “quantum correlated sampling lemma”, a quantum analogue of Holenstein’s classical correlated‑sampling technique. This lemma shows that two non‑communicating parties, each given a description of states |ψ_i⟩ and |ϕ_i⟩ that are close, can locally generate a joint state approximating both using a pre‑shared entangled “embezzlement” resource. This generalizes the universal embezzlement results of van Dam and Hayden to the approximate‑sampling scenario.

Combining multiplicativity of VAL*⁺ with the rounding guarantee yields the main bound
VAL*(G^{⊗k}) ≤ (1‑C·(1‑VAL*(G))^{c})^{k/2}.
Thus the entangled value decays exponentially in the number of repetitions, with a universal exponent. For expanding games the exponent c=1, matching the best known bounds for XOR and unique games.

Beyond parallel repetition, the framework applies to products of non‑identical projection games, giving a unified tool for studying value under arbitrary products. As an application, the authors show that for the class MIP*_1,2 (two‑prover one‑round interactive proofs with entangled provers) the soundness parameter can be amplified to arbitrarily small values without increasing the number of rounds, a result previously unknown.

The paper also discusses broader implications: the vector‑quantum‑strategy formalism and the quantum correlated‑sampling lemma may be useful for problems such as the minimum output entropy conjecture for quantum channels, quantum cryptographic protocol design, and hardness amplification in quantum complexity theory. Overall, the work provides the first general exponential decay result for entangled projection games, bridging a gap between classical and quantum parallel‑repetition theory and introducing powerful new techniques for quantum information theory.