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

Bounding quantum-classical separations for classes of nonlocal games

We bound separations between the entangled and classical values for several classes of nonlocal t-player games. Our motivating question is whether there is a family of t-player XOR games for which the entangled bias is 1 but for which the classical bias goes down to 0, for fixed t. Answering this question would have important consequences in the study of multi-party communication complexity, as a positive answer would imply an unbounded separation between randomized communication complexity with and without entanglement. Our contribution to answering the question is identifying several general classes of games for which the classical bias can not go to zero when the entangled bias stays above a constant threshold. This rules out the possibility of using these games to answer our motivating question. A previously studied set of XOR games, known not to give a positive answer to the question, are those for which there is a quantum strategy that attains value 1 using a so-called Schmidt state. We generalize this class to mod-m games and show that their classical value is always at least 1/m + (m-1)/m t^{1-t}. Secondly, for free XOR games, in which the input distribution is of product form, we show beta(G) >= beta^*(G)^{2^t} where beta(G) and beta^*(G) are the classical and entangled biases of the game respectively. We also introduce so-called line games, an example of which is a slight modification of the Magic Square game, and show that they can not give a positive answer to the question either. Finally we look at two-player unique games and show that if the entangled value is 1-epsilon then the classical value is at least 1-O(sqrt{epsilon log k}) where k is the number of outputs in the game. Our proofs use semidefinite-programming techniques, the Gowers inverse theorem and hypergraph norms

The Hilbertian Tensor Norm and Entangled Two-Prover Games

We study tensor norms over Banach spaces and their relations to quantum information theory, in particular their connection with two-prover games. We consider a version of the Hilbertian tensor norm $\gamma_2$ and its dual $\gamma_2^*$ that allow us to consider games with arbitrary output alphabet sizes. We establish direct-product theorems and prove a generalized Grothendieck inequality for these tensor norms. Furthermore, we investigate the connection between the Hilbertian tensor norm and the set of quantum probability distributions, and show two applications to quantum information theory: firstly, we give an alternative proof of the perfect parallel repetition theorem for entangled XOR games; and secondly, we prove a new upper bound on the ratio between the entangled and the classical value of two-prover games.Comment: 33 pages, some of the results have been obtained independently in arXiv:1007.3043v2, v2: an error in Theorem 4 has been corrected; Section 6 rewritten, v3: completely rewritten in order to improve readability; title changed; references added; published versio

Quantum Proofs

Quantum information and computation provide a fascinating twist on the notion of proofs in computational complexity theory. For instance, one may consider a quantum computational analogue of the complexity class \class{NP}, known as QMA, in which a quantum state plays the role of a proof (also called a certificate or witness), and is checked by a polynomial-time quantum computation. For some problems, the fact that a quantum proof state could be a superposition over exponentially many classical states appears to offer computational advantages over classical proof strings. In the interactive proof system setting, one may consider a verifier and one or more provers that exchange and process quantum information rather than classical information during an interaction for a given input string, giving rise to quantum complexity classes such as QIP, QSZK, and QMIP* that represent natural quantum analogues of IP, SZK, and MIP. While quantum interactive proof systems inherit some properties from their classical counterparts, they also possess distinct and uniquely quantum features that lead to an interesting landscape of complexity classes based on variants of this model. In this survey we provide an overview of many of the known results concerning quantum proofs, computational models based on this concept, and properties of the complexity classes they define. In particular, we discuss non-interactive proofs and the complexity class QMA, single-prover quantum interactive proof systems and the complexity class QIP, statistical zero-knowledge quantum interactive proof systems and the complexity class \class{QSZK}, and multiprover interactive proof systems and the complexity classes QMIP, QMIP*, and MIP*.Comment: Survey published by NOW publisher

Bounding quantum-classical separations for classes of nonlocal games

We bound separations between the entangled and classical values for several classes of nonlocal t-player games. Our motivating question is whether there is a family of t-player XOR games for which the entangled bias is 1 but for which the classical bias goes down to 0, for fixed t. Answering this question would have important consequences in the study of multi-party communication complexity, as a positive answer would imply an unbounded separation between randomized communication complexity with and without entanglement. Our contribution to answering the question is identifying several general classes of games for which the classical bias can not go to zero when the entangled bias stays above a constant threshold. This rules out the possibility of using these games to answer our motivating question. A previously studied set of XOR games, known not to give a positive answer to the question, are those for which there is a quantum strategy that attains value 1 using a so-called Schmidt state. We generalize this class to mod-m games and show that their classical value is always at least 1m+m−1mt1−t. Secondly, for free XOR games, in which the input distribution is of product form, we show β(G)≥β∗(G)2t where β(G) and β∗(G) are the classical and entangled biases of the game respectively. We also introduce so-called line games, an example of which is a slight modification of the Magic Square game, and show that they can not give a positive answer to the question either. Finally we look at two-player unique games and show that if the entangled value is 1−ϵ then the classical value is at least 1−O(√ϵlogk) where k is the number of outputs in the game. Our proofs use semidefinite-programming techniques, the Gowers inverse theorem and hypergraph norms

Entangled Games Are Hard to Approximate

We establish the first hardness results for the problem of computing the value of one-round games played by a verifier and a team of provers 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) a quantum verifier and two entangled provers or (ii) a classical verifier and three entangled provers. 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 of entangled-prover games to within a constant. Using our techniques we also show that every language in PSPACE has a two-prover one-round interactive proof system with perfect completeness and soundness 1-1/poly even against entangled provers. We start our proof by describing two ways to modify classical multiprover games to make them resistant to entangled provers. 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-prover games cannot be computed by semidefinite programs that are polynomial in the size of the verifier's system, a method that has been successful for more restricted quantum games

Survey on nonlocal games and operator space theory

This review article is concerned with a recently uncovered connection between operator spaces, a noncommutative extension of Banach spaces, and quantum nonlocality, a striking phenomenon which underlies many of the applications of quantum mechanics to information theory, cryptography, and algorithms. Using the framework of nonlocal games, we relate measures of the nonlocality of quantum mechanics to certain norms in the Banach and operator space categories. We survey recent results that exploit this connection to derive large violations of Bell inequalities, study the complexity of the classical and quantum values of games and their relation to Grothendieck inequalities, and quantify the nonlocality of different classes of entangled states