Three-Player Entangled XOR Games are NP-Hard to Approximate

We show that for any Є > 0 the problem of finding a factor (2 - Є) approximation to the entangled value of a three-player XOR game is NP-hard. Equivalently, the problem of approximating the largest possible quantum violation of a tripartite Bell correlation inequality to within any multiplicative constant is NP-hard. These results are the first constant-factor hardness of approximation results for entangled games or quantum violations of Bell inequalities shown under the sole assumption that P≠NP. They can be thought of as an extension of Håstad's optimal hardness of approximation results for MAX-E3-LIN2 [J. ACM, 48 (2001), pp. 798--859] to the entangled-player setting. The key technical component of our work is a soundness analysis of a plane-vs-point low-degree test against entangled players. This extends and simplifies the analysis of the multilinearity test by Ito and Vidick [Proceedings of the 53rd FOCS, IEEE, Piscataway, NJ, 2012, pp. 243-252]. Our results demonstrate the possibility of efficient reductions between entangled-player games and our techniques may lead to further hardness of approximation results

Three-player entangled XOR games are NP-hard to approximate

We show that for any ε > 0 the problem of finding a factor (2 - ε) approximation to the entangled value of a three-player XOR game is NP-hard. Equivalently, the problem of approximating the largest possible quantum violation of a tripartite Bell correlation inequality to within any multiplicative constant is NP-hard. These results are the first constant-factor hardness of approximation results for entangled games or quantum violations of Bell inequalities shown under the sole assumption that P≠NP. They can be thought of as an extension of Hástad's optimal hardness of approximation results for MAX-E3-LIN2 (JACM'01) to the entangled-player setting. The key technical component of our work is a soundness analysis of a point-vs-plane low-degree test against entangled players. This extends and simplifies the analysis of the multilinearity test by Ito and Vidick (FOCS'12). Our results demonstrate the possibility for efficient reductions between entangled-player games and our techniques may lead to further hardness of approximation results

Three-Player Entangled XOR Games are NP-Hard to Approximate

We show that for any Є > 0 the problem of finding a factor (2 - Є) approximation to the entangled value of a three-player XOR game is NP-hard. Equivalently, the problem of approximating the largest possible quantum violation of a tripartite Bell correlation inequality to within any multiplicative constant is NP-hard. These results are the first constant-factor hardness of approximation results for entangled games or quantum violations of Bell inequalities shown under the sole assumption that P≠NP. They can be thought of as an extension of Håstad's optimal hardness of approximation results for MAX-E3-LIN2 [J. ACM, 48 (2001), pp. 798--859] to the entangled-player setting. The key technical component of our work is a soundness analysis of a plane-vs-point low-degree test against entangled players. This extends and simplifies the analysis of the multilinearity test by Ito and Vidick [Proceedings of the 53rd FOCS, IEEE, Piscataway, NJ, 2012, pp. 243-252]. Our results demonstrate the possibility of efficient reductions between entangled-player games and our techniques may lead to further hardness of approximation results

Quantum XOR Games

We introduce quantum XOR games, a model of two-player one-round games that extends the model of XOR games by allowing the referee's questions to the players to be quantum states. We give examples showing that quantum XOR games exhibit a wide range of behaviors that are known not to exist for standard XOR games, such as cases in which the use of entanglement leads to an arbitrarily large advantage over the use of no entanglement. By invoking two deep extensions of Grothendieck's inequality, we present an efficient algorithm that gives a constant-factor approximation to the best performance players can obtain in a given game, both in case they have no shared entanglement and in case they share unlimited entanglement. As a byproduct of the algorithm we prove some additional interesting properties of quantum XOR games, such as the fact that sharing a maximally entangled state of arbitrary dimension gives only a small advantage over having no entanglement at all.Comment: 43 page

Algorithms, Bounds, and Strategies for Entangled XOR Games

We study the complexity of computing the commuting-operator value $\omega^*$ of entangled XOR games with any number of players. We introduce necessary and sufficient criteria for an XOR game to have $\omega^* = 1$, and use these criteria to derive the following results: 1. An algorithm for symmetric games that decides in polynomial time whether $\omega^* = 1$ or $\omega^* < 1$, 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 Navascu\'{e}s-Pironio-Ac\'{i}n (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 $\omega^* < 1$, 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 $2(\omega^* - \omega)$ arbitrarily close to the maximum possible value of $1$ (and as a consequence, achieving an unbounded bias ratio), answering an open question of Bri\"{e}t and Vidick. 4. Existence of an unsatisfiable phase for random (non-symmetric) XOR games: that is, we show that there exists a constant $C_k^{\text{unsat}}$ depending only on the number $k$ of players, such that a random $k$-XOR game over an alphabet of size $n$ has $\omega^* < 1$ with high probability when the number of clauses is above $C_k^{\text{unsat}} n$. 5. A lower bound of $\Omega(n \log(n)/\log\log(n))$ 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.Comment: 55 page

Quantum XOR Games

We introduce quantum XOR games, a model of two-player, one-round games that extends the model of XOR games by allowing the referee’s questions to the players to be quantum states. We give examples showing that quantum XOR games exhibit a wide range of behaviors that are known not to exist for standard XOR games, such as cases in which the use of entanglement leads to an arbitrarily large advantage over the use of no entanglement. By invoking two deep extensions of Grothendieck’s inequality, we present an efficient algorithm that gives a constant-factor approximation to the best performance that players can obtain in a given game, both in the case that they have no shared entanglement and that they share unlimited entanglement. As a byproduct of the algorithm, we prove some additional interesting properties of quantum XOR games, such as the fact that sharing a maximally entangled state of arbitrary dimension gives only a small advantage over having no entanglement at all