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, $\omega_q$ is its entangled value, and $\omega_{sdp}$ is the optimum value of the corresponding semidefinite program, then $0.68\,\omega_{sdp} < \omega_q \leq \omega_{sdp}$.
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, ωq is its entangled value, and ωsdp is the optimum value of the corresponding semidefinite program, then 0.68ωsdp<ωq≤ωsdp.
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}.
We prove an explicit upper bound on the amount of entanglement required by any strategy in a two-player cooperative game with classical questions and quantum answers. Specifically, we show that every strategy for a game with n-bit questions and n-qubit answers can be implemented exactly by players who share an entangled state of no more than 5n qubits--a bound which is optimal to within a factor of 5/2. Previously, no upper bound at all was known on the amount of entanglement required even to approximate such a strategy. It follows that the problem of computing the value of these games is in NP, whereas previously this problem was not known to be computable.
In a two-player game, two cooperating but non communicating players, Alice and Bob, receive inputs taken from a probability distribution. Each of them produces an output and they win the game if they satisfy some predicate on their inputs/outputs. The entangled value ω∗(G) of a game G is the maximum probability that Alice and Bob can win the game if they are allowed to share an entangled state prior to receiving their inputs. The n-fold parallel repetition Gn of G consists of n instances of G where Alice and Bob receive all the inputs at the same time and must produce all the outputs at the same time. They win Gn if they win each instance of G. Recently, there has been a series of works showing parallel repetition with exponential decay for projection games [DSV13], games on the uniform distribution [CS14] and for free games, i.e. games on a product distribution [JPY13]. This article is meant to be a follow up of [CS14], where we improve and simplify several parts of our previous paper. Our main result is that for any free game G with value ω∗(G)=1−ε, we have ω∗(Gn)≤(1−ε2)Ω(log(l)n) where l is the size of the output set of the game. This result improves on both the results in [JPY13] and [CS14]. The framework we use can also be extended to free projection games. We show that for a free projection game G with value ω∗(G)=1−ε, we have ω∗(Gn)≤(1−ε)Ω(n).
We study the complexity of computing the commuting-operator value ω∗ of entangled XOR games with any number of players. We introduce necessary and sufficient criteria for an XOR game to have ω∗=1, and use these criteria to derive the following results: 1. An algorithm for symmetric games that decides in polynomial time whether ω∗=1 or ω∗<1, a task that was not previously known to be decidable, together with a simple tensor-product strategy that achieves value 1 in the former case. The only previous candidate algorithm for this problem was the Navascués-Pironio-Acín (also known as noncommutative Sum of Squares or ncSoS) hierarchy, but no convergence bounds were known. 2. A family of games with three players and with ω∗<1, where it takes doubly exponential time for the ncSoS algorithm to witness this (in contrast with our algorithm which runs in polynomial time). 3. A family of games achieving a bias difference 2(ω∗−ω) arbitrarily close to the maximum possible value of 1 (and as a consequence, achieving an unbounded bias ratio), answering an open question of Briët and Vidick. 4. Existence of an unsatisfiable phase for random (non-symmetric) XOR games: that is, we show that there exists a constant Ckunsat depending only on the number k of players, such that a random k-XOR game over an alphabet of size n has ω∗<1 with high probability when the number of clauses is above Ckunsatn. 5. A lower bound of Ω(nlog(n)/loglog(n)) on the number of levels in the ncSoS hierarchy required to detect unsatisfiability for most random 3-XOR games. This is in contrast with the classical case where the n-th level of the sum-of-squares hierarchy is equivalent to brute-force enumeration of all possible solutions.
We establish the first hardness results for the problem of computing the value of one-round games played by a referee and a team of players who can share quantum entanglement. In particular, we show that it is NP-hard to approximate within an inverse polynomial the value of a one-round game with (i) quantum referee and two entangled players or (ii) classical referee and three entangled players. Previously it was not even known if computing the value exactly is NP-hard. We also describe a mathematical conjecture, which, if true, would imply hardness of approximation to within a constant.We start our proof by describing two ways to modify classical multi-player games to make them resistant to entangled players. We then show that a strategy for the modified game that uses entanglement can be "rounded'' to one that does not. The results then follow from classical inapproximability bounds. Our work implies that, unless P = NP, the values of entangled-player games cannot be computed by semidefinite programs that are polynomial in the size of the referee's system, a method that has been successful for more restricted quantum games.
This paper considers a special class of nonlocal games (G,ψ), where G is a two-player one-round game, and ψ is a bipartite state independent of G. In the game (G,ψ), the players are allowed to share arbitrarily many copies of ψ. The value of the game (G,ψ), denoted by ω∗(G,ψ), is the supremum of the winning probability that the players can achieve with arbitrarily many copies of preshared states ψ. For a noisy maximally entangled state ψ, a two-player one-round game G and an arbitrarily small precision ϵ>0, this paper proves an upper bound on the number of copies of ψ for the players to win the game with a probability ϵ close to ω∗(G,ψ). A noisy maximally entangled state is a two-qudit state with both marginals being completely mixed states and the maximal correlation being less than 1. In particular, it includes $(1-\epsilon)|\Psi_m\rangle\langle\Psi_m|+\epsilon\frac{\mathbbm{1}_m}{m}\otimes\frac{\mathbbm{1}_m}{m}$ for ϵ>0, where ∣Ψm⟩=m1∑i=0m−1∣m,m⟩ is an m-dimensional maximally entangled state. Hence, it is feasible to approximately compute ω∗(G,ψ) to an arbitrary precision. Recently, a breakthrough result by Ji et al. showed that it is undecidable to approximate the values of nonlocal games to a constant precision, when the players preshare arbitrarily many copies of perfect maximally entangled states, which implies that MIP∗=RE. In contrast, our result implies the hardness of approximating nonlocal games collapses when the preshared maximally entangled states are noisy. The paper develops a theory of Fourier analysis on matrix spaces by extending a number of techniques in Boolean analysis and Hermitian analysis to matrix spaces. We establish a series of new techniques, such as a quantum invariance principle and a hypercontractive inequality for random operators, which we believe have further applications. (A corrected version is attached.)