A parallel repetition theorem for entangled projection games

Abstract

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 − ε < 1, the value of the k-fold repetition of G goes to zero as O((1 − ε c ) k ), for some universal constant c ≥ 1. If furthermore the constraint graph of G is expanding we obtain the optimal c = 1. Previously exponential decay of the entangled value under parallel repetition was only known for the case of XOR and unique games. To prove the theorem we extend an analytical framework 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, the algorithm 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 to the following approximate scenario: two non-communicating parties, given classical descriptions of bipartite states |ψ , |ϕ respectively such that |ψ ≈ |ϕ , are able to locally generate a joint entangled state |Ψ ≈ |ψ ≈ |ϕ using an initial entangled state that is independent of their inputs.

A Parallel Repetition Theorem for Entangled Projection Games

“…Using the explicit formula for the PGM one can verify that the resulting value exactly corresponds to G ⊗ Id |B 2 2 * , which proves the first inequality in (6). We refer to the full version [39] for details.…”

“…The first relaxation, denoted G 2 * , is related to playing a "squared" version of G with two players Bob and Bob' treated symmetrically. It is defined in Section III-A, and is easily seen to give a good approximation to VAL * , as shown in the following lemma (see [39,Section 3] for the proof):…”

“…We give a self-contained proof of the lemma in [39,Section 3], but readers familiar with quantum information theory may find it interesting to note that a direct proof of (6) can be derived using known properties of the pretty-good measurement (PGM) [30], [31]. We briefly indicate how, before proceeding to give a self-contained proof.…”

“…Finally, in Section IV we state the quantum correlated sampling lemma. For lack of space, almost all proofs are omitted from this extended abstract, but can be found in the full version of this paper available on the arXiv [39].…”

“…For lack of space, the complete proof is omitted from this extended abstract but can be found in [39,Section 4]. The definition of VAL * + is motivated by the following multiplicative property.…”