We show that for any @math there is an XOR game @math with @math inputs for one player and @math inputs for the other player such that @math ebits are required for any strategy achieving bias that is at least a multiplicative factor @math from optimal. This gives an exponential improvement in both the number of inputs or outputs and the noise tolerance of any previously-known self-test for highly entangled states. Up to the exponent @math the scaling of our bound with @math is tight: for any XOR game there is an @math -optimal strategy using @math ebits, irrespective of the number of questions in the game.
These results characterize the dimension of exactly optimal strategies in any XOR game. To the best of our knowledge, even if one considers arbitrary two-player games the @math game remains the most efficient (in terms of total number of inputs and outputs per party) test for @math -qubit maximally entangled states. In particular, although there is strong indication that certain Bell inequalities, such as the @math inequality, have a quantum bound that may only be achieved in the limit of infinite dimensions @cite_9 , no such result has been rigorously proven. There are examples of two-player one round games which provably require infinite-dimensional entanglement in order to be played optimally @cite_14 @cite_3 , but these require the exchange of quantum messages between the referee and the players. Recently Slofstra @cite_10 showed the existence of a game for which a value @math can be attained using infinite-dimensional commuting-operator strategies, but it is not known if there exists a tensor product strategy achieving this value; in particular there is no optimal entangled state'' for this game.
@cite_9 @cite_14 @cite_10 @cite_3
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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. We give an explicit family of XOR games with O(n)-bit questions requiring 2n ebits to play near-optimally. More generally, we introduce a new technique for proving lower bounds on the amount of entanglement required by an XOR game: we show that near-optimal strategies for an XOR game G correspond to approximate representations of a certain C*-algebra associated to G. Our results extend an earlier theorem of Tsirelson characterising the set of quantum strategies which implement extremal quantum correlations. Bell inequality violations correspond to behavior of entangled quantum systems that cannot be simulated classically. We give two new two-player games with Bell inequality violations that are stronger, fully explicit, and arguably simpler than earlier work.The first game is based on the Hidden Matching problem of quantum communication complexity, introduced by Bar-Yossef, Jayram, and Kerenidis. This game can be won with probability 1 by a quantum strategy using a maximally entangled state with local dimension n (e.g., log n EPR-pairs), while we show that the winning probability of any classical strategy differs from 1 2 by at most O(log(n) sqrt(n)).The second game is based on the integrality gap for Unique Games by Khot and Vishnoi and the quantum rounding procedure of Kempe, Regev, and Toner. Here n-dimensional entanglement allows to win the game with probability 1 (log n)^2, while the best winning probability without entanglement is 1 n. This near-linear ratio ( Bell inequality violation'') is near-optimal, both in terms of the local dimension of the entangled state, and in terms of the number of possible outputs of the two players. We construct a linear system non-local game which can be played perfectly using a limit of finite-dimensional quantum strategies, but which cannot be played perfectly on any finite-dimensional Hilbert space, or even with any tensor-product strategy. In particular, this shows that the set of (tensor-product) quantum correlations is not closed. The constructed non-local game provides another counterexample to the "middle" Tsirelson problem, with a shorter proof than our previous paper (though at the loss of the universal embedding theorem). We also show that it is undecidable to determine if a linear system game can be played perfectly with a finite-dimensional strategy, or a limit of finite-dimensional quantum strategies. We study a model of communication complexity that encompasses many well-studied problems, including classical and quantum communication complexity, the complexity of simulating distributions arising from bipartite measurements of shared quantum states, and XOR games. In this model, Alice gets an input x, Bob gets an input y, and their goal is to each produce an output a, b distributed according to some pre-specified joint distribution p(a, b|x, y). Our results apply to any non-signaling distribution, that is, those where Alice's marginal distribution does not depend on Bob's input, and vice versa. By taking a geometric view of the non-signaling distributions, we introduce a simple new technique based on affine combinations of lower-complexity distributions, and we give the first general technique to apply to all these settings, with elementary proofs and very intuitive interpretations. Specifically, we introduce two complexity measures, one which gives lower bounds on classical communication, and one for quantum communication. These measures can be expressed as convex optimization problems. We show that the dual formulations have a striking interpretation, since they coincide with maximum violations of Bell and Tsirelson inequalities. The dual expressions are closely related to the winning probability of XOR games. Despite their apparent simplicity, these lower bounds subsume many known communication complexity lower bound methods, most notably the recent lower bounds of Linial and Shraibman for the special case of Boolean functions. We show that as in the case of Boolean functions, the gap between the quantum and classical lower bounds is at most linear in the size of the support of the distribution, and does not depend on the size of the inputs. This translates into a bound on the gap between maximal Bell and Tsirelson inequality violations, which was previously known only for the case of distributions with Boolean outcomes and uniform marginals. It also allows us to show that for some distributions, information theoretic methods are necessary to prove strong lower bounds. Finally, we give an exponential upper bound on quantum and classical communication complexity in the simultaneous messages model, for any non-signaling distribution. One consequence of this is a simple proof that any quantum distribution can be approximated with a constant number of bits of communication. This paper investigates various aspects of the nonlocal effects that can arise when entangled quantum information is shared between two parties. A natural framework for studying nonlocality is that of cooperative games with incomplete information, where two cooperating players may share entanglement. Here, nonlocality can be quantified in terms of the values of such games. We review some examples of non-locality and show that it can profoundly affect the soundness of two-prover interactive proof systems. We then establish limits on nonlocal behavior by upper-bounding the values of several of these games. These upper bounds can be regarded as generalizations of the so-called Tsirelson inequality. We also investigate the amount of entanglement required by optimal and nearly optimal quantum strategies. We study the problem of approximating the commuting-operator value of a two-player non-local game. It is well-known that it is @math -complete to decide whether the classical value of a non-local game is 1 or @math . Furthermore, as long as @math is small enough, this result does not depend on the gap @math . In contrast, a recent result of Fitzsimons, Ji, Vidick, and Yuen shows that the complexity of computing the quantum value grows without bound as the gap @math decreases. In this paper, we show that this also holds for the commuting-operator value of a game. Specifically, in the language of multi-prover interactive proofs, we show that the power of @math (proofs with two provers, one round, completeness probability @math , soundness probability @math , and commuting-operator strategies) can increase without bound as the gap @math gets arbitrarily small. Our results also extend naturally in two ways, to perfect zero-knowledge protocols, and to lower bounds on the complexity of computing the approximately-commuting value of a game. Thus we get lower bounds on the complexity class @math - @math of perfect zero-knowledge multi-prover proofs with approximately-commuting operator strategies, as the gap @math gets arbitrarily small. While we do not know any computable time upper bound on the class @math , a result of the first author and Vidick shows that for @math and @math , the class @math , with constant communication from the provers, is contained in @math . We give a lower bound of @math (ignoring constants inside the function) for this class, which is tight up to polynomial factors assuming the exponential time hypothesis. We show that it is NP-hard to approximate, to within an additive constant, the maximum success probability of players sharing quantum entanglement in a two-player game with classical questions of logarithmic length and classical answers of constant length. As a corollary, the inclusion NEXP ⊆ MIP*, first shown by Ito and Vidick (FOCS'12) with three provers, holds with two provers only. The proof is based on a simpler, improved analysis of the low-degree test of Raz and Safra (STOC'97) against two entangled provers. We propose a simple yet rich model to extend strategic games to the quantum setting, in which we define quantum Nash and correlated equilibria and study the relations between classical and quantum equilibria. Unlike all previous work that focused on qualitative questions on specific games of very small sizes, we quantitatively address the following fundamental question for general games of growing sizes: How much "advantage" can playing quantum strategies provide, if any? Two measures of the advantage are studied. 1. Since game mainly is about each player trying to maximize individual payoff, a natural measure is the increase of payoff by playing quantum strategies. We consider natural mappings between classical and quantum states, and study how well those mappings preserve equilibrium properties. Among other results, we exhibit a correlated equilibrium p whose quantum superposition counterpart [EQUATION] is far from being a quantum correlated equilibrium; actually a player can increase her payoff from almost 0 to almost 1 in a [0, 1]-normalized game. We achieve this by a tensor product construction on carefully designed base cases. The result can also be interpreted as in Meyer's comparison [47]: In a state no classical player can gain, one player using quantum computers has an huge advantage than continuing to play classically. 2. Another measure is the hardness of generating correlated equilibria, for which we propose to study correlation complexity, a new complexity measure for correlation generation. We show that there are n-bit correlated equilibria which can be generated by only one EPR pair followed by local operation (without communication), but need at least log2(n) classical shared random bits plus communication. The randomized lower bound can be improved to n, the best possible, assuming (even a much weaker version of) a recent conjecture in linear algebra. We believe that the correlation complexity, as a complexity-theoretical counterpart of the celebrated Bell's inequality, has independent interest in both physics and computational complexity theory and deserves more explorations. In this paper we show that sufficient multi-partite quantum entanglement helps in fair and unbiased election of a leader in a distributed network of processors with only linear classical communication complexity. We show that a total of @math distinct multi-partite maximally entanglement sets (ebits) are capable of supporting such a protocol in the presence of nodes that may lie and thus be biased. Here, @math is the number of nodes in the network. We also demonstrate the difficulty of performing unbiased and fair election of a leader with linear classical communication complexity in the absence of quantum entanglement even if all nodes have perfect random bit generators. We show that the presence of a sufficient number @math of biased agents leads to a non-zero limiting probability of biased election of the leader, whereas, the presence of a smaller number @math of biased agents matters little. We define two new related complexity classes motivated by the our leader election problem and discuss a few open questions. By introducing a quantitative degree of commutativity'' in terms of the angle between spin observables we present two tight quantitative trade-off relations in the case of two qubits. First, for entangled states, between the degree of commutativity of local observables and the maximal amount of violation of the Bell inequality: if both local angles increase from zero to @math (i.e., the degree of local commutativity decreases), the maximum violation of the Bell inequality increases. Secondly, a converse trade-off relation holds for separable states: if both local angles approach @math the maximal value obtainable for the correlations in the Bell inequality decreases and thus the non-violation increases. As expected, the extremes of these relations are found in the case of anticommuting local observables where, respectively, the bounds of @math and @math hold for the expectation value of the Bell operator. The trade-off relations show that noncommmutativity gives a more than classical result'' for entangled states, whereas a less than classical result'' is obtained for separable states. The experimental relevance of the trade-off relation for separable states is that it provides an experimental test for two qubit entanglement. Its advantages are twofold: in comparison to violations of Bell inequalities it is a stronger criterion and in comparison to entanglement witnesses it needs to make less strong assumptions about the observables implemented in the experiment. We show that any language in nondeterministic time @math , where the number of iterated exponentials is an arbitrary function @math , can be decided by a multiprover interactive proof system with a classical polynomial-time verifier and a constant number of quantum entangled provers, with completeness @math and soundness @math , where the number of iterated exponentials is @math and @math is a universal constant. The result was previously known for @math and @math ; we obtain it for any time-constructible function @math . The result is based on a compression technique for interactive proof systems with entangled provers that significantly simplifies and strengthens a protocol compression result of Ji (STOC'17). As a separate consequence of this technique we obtain a different proof of Slofstra's recent result (unpublished) on the uncomputability of the entangled value of multiprover games. Finally, we show that even minor improvements to our compression result would yield remarkable consequences in computational complexity theory and the foundations of quantum mechanics: first, it would imply that the class MIP* contains all computable languages; second, it would provide a negative resolution to a multipartite version of Tsirelson's problem on the relation between the commuting operator and tensor product models for quantum correlations. 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 I sub 3322 inequality is the simplest bipartite two-outcome Bell inequality beyond the Clauser-Horne-Shimony-Holt (CHSH) inequality, consisting of three two-outcome measurements per party. In the case of the CHSH inequality the maximal quantum violation can already be attained with local two-dimensional quantum systems; however, there is no such evidence for the I sub 3322 inequality. In this paper a family of measurement operators and states is given which enables us to attain the maximum quantum value in an infinite-dimensional Hilbert space. Further, it is conjectured that our construction is optimal in the sense that measuring finite-dimensional quantum systems is not enough to achieve the true quantum maximum. We also describe an efficient iterative algorithm for computing quantum maximum of an arbitrary two-outcome Bell inequality in any given Hilbert space dimension. This algorithm played a key role in obtaining our results for the I sub 3322 inequality, and we also applied it to improve on our previous results concerning the maximum quantum violation of several bipartite two-outcome Bell inequalities with up to five settings per party. We consider two-player games played for an infinite number of rounds, with ω-regular winning conditions. The games may be concurrent, in that the players choose their moves simultaneously and independently, and probabilistic, in that the moves determine a probability distribution for the successor state. We introduce quantitative game µ-calculus, and we show that the maximal probability of winning such games can be expressed as the fixpoint formulas in this calculus. We develop the arguments both for deterministic and for probabilistic concurrent games; as a special case, we solve probabilistic turn-based games with ω-regular winning conditions, which was also open. We also characterize the optimality, and the memory requirements, of the winning strategies. In particular, we show that while memoryless strategies suffice for winning games with safety and reachability conditions, Buchi conditions require the use of strategies with infinite memory. The existence of optimal strategies, as opposed to e-optimal, is only guaranteed in games with safety winning conditions. Tsirelson's problem asks whether the commuting operator model for two-party quantum correlations is equivalent to the tensor-product model. We give a negative answer to this question by showing that there are non-local games which have perfect commuting-operator strategies, but do not have perfect tensor-product strategies. The weak Tsirelson problem, which is known to be equivalent to Connes embedding problem, remains open. The examples we construct are instances of (binary) linear system games. For such games, previous results state that the existence of perfect strategies is controlled by the solution group of the linear system. Our main result is that every finitely-presented group embeds in some solution group. As an additional consequence, we show that the problem of determining whether a linear system game has a perfect commuting-operator strategy is undecidable. A quantum algorithm for an oracle problem can be understood as a quantum strategy for a player in a two-player zero-sum game in which the other player is constrained to play classically. I formalize this correspondence and give examples of games (and hence oracle problems) for which the quantum player can do better than would be possible classically. The most remarkable example is the Bernstein-Vazirani quantum search algorithm which I show creates no entanglement at any timestep. We consider two-player games played over finite state spaces for an infinite number of rounds. At each state, the players simultaneously choose moves; the moves determine a successor state. It is often advantageous for players to choose probability distributions over moves, rather than single moves. Given a goal, for example, reach a target state, the question of winning is thus a probabilistic one: what is the maximal probability of winning from a given state? On these game structures, two fundamental notions are those of equivalences and metrics. Given a set of winning conditions, two states are equivalent if the players can win the same games with the same probability from both states. Metrics provide a bound on the difference in the probabilities of winning across states, capturing a quantitative notion of state similarity. We introduce equivalences and metrics for two-player game structures, and we show that they characterize the difference in probability of winning games whose goals are expressed in the quantitative mu-calculus. The quantitative mu-calculus can express a large set of goals, including reachability, safety, and omega-regular properties. Thus, we claim that our relations and metrics provide the canonical extensions to games, of the classical notion of bisimulation for transition systems. We develop our results both for equivalences and metrics, which generalize bisimulation, and for asymmetrical versions, which generalize simulation. We show that any number of parties can coherently exchange any one pure quantum state for another, without communication, given prior shared entanglement. Two applications of this fact to the study of multi-prover quantum interactive proof systems are given. First, we prove that there exists a one-round two-prover quantum interactive proof system for which no finite amount of shared entanglement allows the provers to implement an optimal strategy. More specifically, for every fixed input string, there exists a sequence of strategies for the provers, with each strategy requiring more entanglement than the last, for which the probability for the provers to convince the verifier to accept approaches 1. It is not possible, however, for the provers to convince the verifier to accept with certainty with a finite amount of shared entanglement. The second application is a simple proof that multi-prover quantum interactive proofs can be transformed to have near-perfect completeness by the addition of one round of communication. A proof of Bell's theorem using two maximally entangled states of two qubits is presented. It exhibits a similar logical structure to Hardy's argument of nonlocality without inequalities''. However, it works for 100 of the runs of a certain experiment. Therefore, it can also be viewed as a Greenberger-Horne-Zeilinger-like proof involving only two spacelike separated regions.