A Lower Bound on the Value of Entangled Binary Games

TL;DR

This paper establishes a semidefinite programming method to efficiently compute or bound the entangled value of binary games, including uniform and general cases, providing a new lower bound for the latter.

Contribution

It extends the class of binary games with efficiently computable entangled values and introduces a novel lower bound for general two-player games based on output set size.

Findings

Entangled XOR-games' values can be computed via semidefinite programs.

Uniform binary games' entangled value is efficiently computable.

A new lower bound (0.68 times SDP value) for general binary games' entangled value.

Abstract

A two-player one-round binary game consists of two cooperative players who each replies by one bit to a message that he receives privately; they win the game if both questions and answers satisfy some predetermined property. A game is called entangled if the players are allowed to share a priori entanglement. It is well-known that the maximum winning probability (value) of entangled XOR-games (binary games in which the predetermined property depends only on the XOR of the two output bits) can be computed by a semidefinite program. In this paper we extend this result in the following sense; if a binary game is uniform, meaning that in an optimal strategy the marginal distributions of the output of each player are uniform, then its entangled value can be efficiently computed by a semidefinite program. We also introduce a lower bound on the entangled value of a general two-player one-round game; this bound depends on the size of the output set of each player and can be computed by a semidefinite program. In particular, we show that if the game is binary, w_q is its entangled value, and w_{sdp} is the optimum value of the corresponding semidefinite program, then 0.68w_{sdp} < w_q <= w_{sdp}.