Characterization of Binary Constraint System Games

We consider a class of nonlocal games that are related to binary constraint systems (BCSs) in a manner similar to the games implicit in the work of Mermin [N.D. Mermin, "Simple unified form for the major no-hidden-variables theorems," Phys. Rev. Lett., 65(27):3373-3376, 1990], but generalized to n binary variables and m constraints. We show that, whenever there is a perfect entangled protocol for such a game, there exists a set of binary observables with commutations and products similar to those exhibited by Mermin. We also show how to derive upper bounds strictly below 1 for the the maximum entangled success probability of some BCS games. These results are partial progress towards a larger project to determine the computational complexity of deciding whether a given instance of a BCS game admits a perfect entangled strategy or not.Comment: Revised version corrects an error in the previous version of the proof of Theorem 1 that arises in the case of POVM measurement

Quantum and non-signalling graph isomorphisms

We introduce the (G,H)-isomorphism game, a new two-player non-local game that classical players can win with certainty iff the graphs G and H are isomorphic. We then define quantum and non-signalling isomorphisms by considering perfect quantum and non-signalling strategies for this game. We prove that non-signalling isomorphism coincides with fractional isomorphism, giving the latter an operational interpretation. We show that quantum isomorphism is equivalent to the feasibility of two polynomial systems obtained by relaxing standard integer programs for graph isomorphism to Hermitian variables. Finally, we provide a reduction from linear binary constraint system games to isomorphism games. This reduction provides examples of quantum isomorphic graphs that are not isomorphic, implies that the tensor product and commuting operator frameworks result in different notions of quantum isomorphism, and proves that both relations are undecidable.Peer ReviewedPostprint (author's final draft

Bell nonlocality

Bell's 1964 theorem, which states that the predictions of quantum theory cannot be accounted for by any local theory, represents one of the most profound developments in the foundations of physics. In the last two decades, Bell's theorem has been a central theme of research from a variety of perspectives, mainly motivated by quantum information science, where the nonlocality of quantum theory underpins many of the advantages afforded by a quantum processing of information. The focus of this review is to a large extent oriented by these later developments. We review the main concepts and tools which have been developed to describe and study the nonlocality of quantum theory, and which have raised this topic to the status of a full sub-field of quantum information science.Comment: 65 pages, 7 figures. Final versio

Generalized Tsirelson Inequalities, Commuting-Operator Provers, and Multi-Prover Interactive Proof Systems

A central question in quantum information theory and computational complexity is how powerful nonlocal strategies are in cooperative games with imperfect information, such as multi-prover interactive proof systems. This paper develops a new method for proving limits of nonlocal strategies that make use of prior entanglement among players (or, provers, in the terminology of multi-prover interactive proofs). Instead of proving the limits for usual isolated provers who initially share entanglement, this paper proves the limits for "commuting-operator provers", who share private space, but can apply only such operators that are commutative with any operator applied by other provers. Commuting-operator provers are at least as powerful as usual isolated but prior-entangled provers, and thus, limits for commuting-operator provers immediately give limits for usual entangled provers. Using this method, we obtain an n-party generalization of the Tsirelson bound for the Clauser-Horne- Shimony-Holt inequality for every n. Our bounds are tight in the sense that, in every n-party case, the equality is achievable by a usual nonlocal strategy with prior entanglement. We also apply our method to a 3-prover 1-round binary interactive proof for NEXP. Combined with the technique developed by Kempe, Kobayashi, Matsumoto, Toner and Vidick to analyze the soundness of the proof system, it is proved to be NP-hard to distinguish whether the entangled value of a 3-prover 1-round binary-answer game is equal to 1 or at most 1-1/p(n) for some polynomial p, where n is the number of questions. This is in contrast to the 2-prover 1-round binary-answer case, where the corresponding problem is efficiently decidable. Alternatively, NEXP has a 3-prover 1-round binary interactive proof system with perfect completeness and soundness 1-2^{-poly}.Comment: 20 pages. v2: An incorrect statement in the abstract about the two-party case is corrected. Relation between this work and a preliminary work by Sun, Yao and Preda is clarifie

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

Noncommutative Nullstellens\"atze and Perfect Games

The foundations of classical Algebraic Geometry and Real Algebraic Geometry are the Nullstellensatz and Positivstellensatz. Over the last two decades the basic analogous theorems for matrix and operator theory (noncommutative variables) have emerged. This paper concerns commuting operator strategies for nonlocal games, recalls NC Nullstellensatz which are helpful, extends these, and applies them to a very broad collection of games. In the process it brings together results spread over different literatures, hence rather than being terse, our style is fairly expository. The main results of this paper are two characterizations, based on Nullstellensatz, which apply to games with perfect commuting operator strategies. The first applies to all games and reduces the question of whether or not a game has a perfect commuting operator strategy to a question involving left ideals and sums of squares. Previously, Paulsen and others translated the study of perfect synchronous games to problems entirely involving a $*$-algebra.The characterization we present is analogous, but works for all games. The second characterization is based on a new Nullstellensatz we derive in this paper. It applies to a class of games we call torically determined games, special cases of which are XOR and linear system games. For these games we show the question of whether or not a game has a perfect commuting operator strategy reduces to instances of the subgroup membership problem and, for linear systems games, we further show this subgroup membership characterization is equivalent to the standard characterization of perfect commuting operator strategies in terms of solution groups. Both the general and torically determined games characterizations are amenable to computer algebra techniques, which we also develop.Comment: 58 page

Near-optimal quantum strategies for nonlocal games, approximate representations, and BCS algebras

Quantum correlations can be viewed as particular abstract states on the tensor product of operator systems which model quantum measurement scenarios. In the paradigm of nonlocal games, this perspective illustrates a connection between optimal strategies and certain representations of a finitely presented $*$-algebra affiliated with the nonlocal game. This algebraic interpretation of quantum correlations arising from nonlocal games has been valuable in recent years. In particular, the connection between representations and strategies has been useful for investigating and separating the various frameworks for quantum correlation as well as in developing cryptographic primitives for untrusted quantum devices. However to make use of this correspondence in a realistic setting one needs mathematical guarantees that this correspondence is robust to noise. We address this issue by considering the situation where the correlations are not ideal. We show that near-optimal finite-dimensional quantum strategies using arbitrary quantum states are approximate representations of the affiliated nonlocal game algebra for synchronous, boolean constraint systems (BCS), and XOR nonlocal games. This result robustly extends the correspondence between optimal strategies and finite-dimensional representations of the nonlocal game algebras for these prominent classes of nonlocal games. We also show that finite-dimensional approximate representations of these nonlocal game algebras are close to near-optimal strategies employing a maximally entangled state. As a corollary, we deduce that near-optimal quantum strategies are close to a near-optimal quantum strategy using a maximally entangled state. A boolean constraint system $B$ is $pp$-definable from another boolean constraint system $B'$ if there is a $pp$-formula defining $B$ over $B'$. There is such a $pp$-formula if all the constraints in $B$ can be defined via conjunctions of relations in $B'$ using additional boolean variables if needed. We associate a finitely presented $*$-algebra, called a BCS algebra, to each boolean constraint system $B$. We show that $pp$-definability can be interpreted algebraically as $*$-homomorphisms between BCS algebras. This allows us to classify boolean constraint languages and separations between various generalized notions of satisfiability. These types of satisfiability are motivated by nonlocal games and the various frameworks for quantum correlations and state-independent contextuality. As an example, we construct a BCS that is $C^*$-satisfiable in the sense that it has a representation on a Hilbert space $H$ but has no tracial representations, and thus no interpretation in terms of commuting operator correlations