XOR Games Research Articles

On the power of quantum entanglement in multipartite quantum XOR games

AbstractWe show that, given , there exist ‐player quantum XOR games for which the entangled bias can be arbitrarily larger than the bias of the game when the players are restricted to separable strategies. In particular, quantum entanglement can be a much more powerful resource than local operations and classical communication to play these games. This result shows a strong contrast to the bipartite case, where it was recently proved that, as a consequence of a noncommutative version of Grothendieck theorem, the entangled bias is always upper bounded by a universal constant times the one‐way classical communication bias. In this sense, our main result can be understood as a counterexample to an extension of such a noncommutative Grothendieck theorem to multilinear forms.

Strength of the non-locality of two-qubit entangled states and its applications

Non-locality is a feature of quantum mechanics that cannot be explained by local realistic theory. It can be detected by the violation of Bell’s inequality. In this work, we have considered the evaluation of Bell’s inequality with the help of the XOR game. In the XOR game, a two-qubit entangled state is shared between the two distant players. It may generate a non-local correlation between the players which contributes to the maximum probability of winning of the game. We have aimed to determine the strength of the non-locality through XOR game. Thus, we have defined a quantity S NL called the strength of non-locality, purely on the basis of the maximum probability of winning of the XOR game. We have also derived the relation between the introduced quantity S NL and the quantity M introduced in [R Horodecki, P Horodecki, and M Horodecki, (1995) Phys. Lett. A 200, 340.], to study the non-locality of a two-qubit entangled state problem in depth. The quantity M may be defined as the sum of the two largest eigenvalues of the correlation matrix of the given entangled state and it determines whether the given entangled state under probe is non-local. Further, we have explored the non-locality of any two-qubit entangled state, whose non-locality cannot be detected by the CHSH inequality. Interestingly, we have found that the newly defined quantity S NL fails to detect non-locality for the entangled state, when the witness operator corresponding to CHSH operator cannot detect the entangled state. To overcome this problem, we have modified the definition of the strength of non-locality and have shown that the modified definition may detect the non-locality of such entangled states, which were earlier undetected by S NL . Furthermore, we have provided two applications of the strength S NL of the non-locality such as (i) establishment of a link between the two-qubit non-locality determined by S NL and the three-qubit non-locality determined by the Svetlichny operator and (ii) determination of the upper bound of the power of the controller in terms of S NL in controlled quantum teleportation.

3XOR Games with Perfect Commuting Operator Strategies Have Perfect Tensor Product Strategies and are Decidable in Polynomial Time

We consider 3XOR games with perfect commuting operator strategies. Given any 3XOR game, we show existence of a perfect commuting operator strategy for the game can be decided in polynomial time. Previously this problem was not known to be decidable. Our proof leads to a construction, showing a 3XOR game has a perfect commuting operator strategy iff it has a perfect tensor product strategy using a 3 qubit (8 dimensional) GHZ state. This shows that for perfect 3XOR games the advantage of a quantum strategy over a classical strategy (defined by the quantum-classical bias ratio) is bounded. This is in contrast to the general 3XOR case where the optimal quantum strategies can require high dimensional states and there is no bound on the quantum advantage. To prove these results, we first show equivalence between deciding the value of an XOR game and solving an instance of the subgroup membership problem on a class of right angled Coxeter groups. We then show, in a proof that consumes most of this paper, that the instances of this problem corresponding to 3XOR games can be solved in polynomial time.

Noncommutative Nullstellensätze 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 literature studies, 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. For context, we mention that Positivstellensätze are behind the standard NPA upper bound on the score players can achieve for a game using a commuting operator strategy. This paper develops analogous NC real algebraic geometry which bears on perfect games.

On the relation between completely bounded and (1,cb)-summing maps with applications to quantum XOR games

In this work we show that, given a linear map from a general operator space into the dual of a C⁎-algebra, its completely bounded norm is upper bounded by a universal constant times its (1,cb)-summing norm. This problem is motivated by the study of quantum XOR games in the field of quantum information theory. In particular, our results imply that for such games entangled strategies cannot be arbitrarily better than those strategies using one-way classical communication.

Quantum one-way versus classical two-way communication in XOR games

In this work, we give an example of exponential separation between quantum and classical resources in the setting of XOR games assisted with communication. Specifically, we show an example of a XOR game for which O(n) bits of two-way classical communication are needed in order to achieve the same value as can be attained with $$\log n$$ qubits of one-way communication. We also find a characterization for the value of a XOR game assisted with a limited amount of two-way communication in terms of tensor norms of normed spaces.

Quantum strategies for simple two-player XOR games

The non-local game scenario provides a powerful framework to study the limitations of classical and quantum correlations, by studying the upper bounds of the winning probabilities those correlations offer in cooperation games where communication between players is prohibited. Building upon results presented in the seminal work of Cleve et al. [1], a straightforward construction to compute the Tsirelson bounds for simple 2-player XOR games is presented. The construction is applied explicitly to some examples, including the Entanglement Assisted Orientation in Space (EAOS) game of Brukner et al. [2], proving for the first time that their proposed quantum strategy is in fact the optimal, as it reaches the Tsirelson bound.

Universal Gaps for XOR Games from Estimates on Tensor Norm Ratios

We define and study XOR games in the framework of general probabilistic theories, which encompasses all physical models whose predictive power obeys minimal requirements. The bias of an XOR game under local or global strategies is shown to be given by a certain injective or projective tensor norm, respectively. The intrinsic (i.e. model-independent) advantage of global over local strategies is thus connected to a universal function r(n, m) called ‘projective–injective ratio’. This is defined as the minimal constant rho such that Vert cdot Vert _{Xotimes _pi Y}leqslant rho ,Vert cdot Vert _{Xotimes _varepsilon Y} holds for all Banach spaces of dimensions dim X=n and dim Y=m, where Xotimes _pi Y and X otimes _varepsilon Y are the projective and injective tensor products. By requiring that X=Y, one obtains a symmetrised version of the above ratio, denoted by r_s(n). We prove that r(n,m)geqslant 19/18 for all n,mgeqslant 2, implying that injective and projective tensor products are never isometric. We then study the asymptotic behaviour of r(n, m) and r_s(n), showing that, up to log factors: r_s(n) is of the order sqrt{n} (which is sharp); r(n, n) is at least of the order n^{1/6}; and r(n, m) grows at least as min {n,m}^{1/8}. These results constitute our main contribution to the theory of tensor norms. In our proof, a crucial role is played by an ‘ell _1/ell _2/ell _{infty } trichotomy theorem’ based on ideas by Pisier, Rudelson, Szarek, and Tomczak-Jaegermann. The main operational consequence we draw is that there is a universal gap between local and global strategies in general XOR games, and that this grows as a power of the minimal local dimension. In the quantum case, we are able to determine this gap up to universal constants. As a corollary, we obtain an improved bound on the scaling of the maximal quantum data hiding efficiency against local measurements.

All tight correlation Bell inequalities have quantum violations

It is by now well-established that there exist non-local games for which the best entanglement-assisted performance is not better than the best classical performance. Here we show in contrast that any two-player XOR game, for which the corresponding Bell inequality is tight, has a quantum advantage. In geometric terms, this means that any correlation Bell inequality for which the classical and quantum maximum values coincide, does not define a facet, i.e. a face of maximum dimension, of the local Bell polytope. Indeed, using semidefinite programming duality, we prove upper bounds on the dimension of these faces, bounding it far away from the maximum. In the special case of non-local computation games, it had been shown before that they are not facet-defining; our result generalises and improves this. As a by-product of our analysis, we find a similar upper bound on the dimension of the faces of the convex body of quantum correlation matrices, showing that (except for the trivial ones expressing the non-negativity of probability) it does not have facets.

Erratum: Three-Player Entangled XOR Games are NP-hard to Approximate

This note indicates an error in the proof of Theorem 3.1 in [T. Vidick, SIAM J. Comput., 45 (2016), pp. 1007--1063]. Due to an induction step in the soundness analysis not being carried out correct...

Steering is an essential feature of non-locality in quantum theory

A physical theory is called non-local when observers can produce instantaneous effects over distant systems. Non-local theories rely on two fundamental effects: local uncertainty relations and steering of physical states at a distance. In quantum mechanics, the former one dominates the other in a well-known class of non-local games known as XOR games. In particular, optimal quantum strategies for XOR games are completely determined by the uncertainty principle alone. This breakthrough result has yielded the fundamental open question whether optimal quantum strategies are always restricted by local uncertainty principles, with entanglement-based steering playing no role. In this work, we provide a negative answer to the question, showing that both steering and uncertainty relations play a fundamental role in determining optimal quantum strategies for non-local games. Our theoretical findings are supported by an experimental implementation with entangled photons.

Entanglement of approximate quantum strategies in XOR games

We characterize the amount of entanglement that is sufficient to play any XOR game near-optimally. We show that for any XOR game G and \eps>0 there is an \eps-optimal strategy for G using \lceil \eps^{-1} \rceil ebits of entanglement, irrespective of the number of questions in the game. By considering the family of XOR games CHSH(n) introduced by Slofstra (Jour. Math. Phys. 2011), we show that this bound is nearly tight: for any \eps>0 there is an n=\Theta(\eps^{-1/5}) such that \Omega(\eps^{-1/5}) ebits are required for any strategy achieving bias that is at least a multiplicative factor (1-\eps) from optimal in CHSH(n).

Classical versus quantum communication in XOR games

We introduce an intermediate setting between quantum nonlocality and communication complexity problems. More precisely, we study the value of XOR games when Alice and Bob are allowed to use a limited amount (c bits) of one-way classical communication compared to their value when they are allowed to use the same amount of one-way quantum communication (c qubits). The key quantity here is the ratio between the quantum and classical value. We provide a universal way to obtain Bell inequality violations of general Bell functionals from XOR games for which the previous quotient is larger than 1. This allows, in particular, to find (unbounded) Bell inequality violations from communication complexity problems in the same spirit as the recent work by Buhrman et al. (PNAS 113(12):3191–3196, 2016). We also provide an example of a XOR game for which the previous quotient is optimal (up to a logarithmic factor) in terms of the amount of information c. Interestingly, this game has only polynomially many inputs per player. For the related problem of separating the classical versus quantum communication complexity of a function, the known examples attaining exponential separation require exponentially many inputs per party.

Better protocol for XOR game using communication protocol and nonlocal boxes

Buhrman showed that an efficient communication protocol implies a reliable XOR game protocol. This idea rederives Linial and Shraibman’s lower bound of randomized and quantum communication complexities, which was derived by using factorization norms, with worse constant factor in much more intuitive way. In this work, we improve and generalize Buhrman’s idea, and obtain a class of lower bounds for randomized communication complexity including an exact Linial and Shraibman’s lower bound as a special case. In the proof, we explicitly construct a protocol for XOR game from a randomized communication protocol by using a concept of nonlocal boxes and Paw lowski et al.’s elegant protocol, which was used for showing the violation of information causality in superquantum theories.

Connes’ embedding problem and winning strategies for quantum XOR games

We consider quantum XOR games, defined in the work of Regev and Vidick [ACM Trans. Comput. Theory 7, 43 (2015)], from the perspective of unitary correlations defined in the work of Harris and Paulsen [Integr. Equations Oper. Theory 89, 125 (2017)]. We show that the winning bias of a quantum XOR game in the tensor product model (respectively, the commuting model) is equal to the norm of its associated linear functional on the unitary correlation set from the appropriate model. We show that Connes’ embedding problem has a positive answer if and only if every quantum XOR game has entanglement bias equal to the commuting bias. In particular, the embedding problem is equivalent to determining whether every quantum XOR game G with a winning strategy in the commuting model also has a winning strategy in the approximate finite-dimensional model.

The structure of nearly-optimal quantum strategies for the non-local XOR games

We consider the infinite family of non-local games CHSH(n). We consider nearly-optimal strategies for CHSH(n). We introduce a notion of approximate homomorphism for strategies and show that any nearly-optimal strategy for CHSH(n) is approximately homomorphic to the canonical optimal CHSH(n) strategy. This demonstrates that any nearly-optimal CHSH(n) strategy must approximately contain the algebraic structure of the canonical optimal strategy.

Linear game non-contextuality and Bell inequalities—a graph-theoretic approach

We study the classical and quantum values of a class of one- and two-party unique games, that generalizes the well-known XOR games to the case of non-binary outcomes. In the bipartite case the generalized XOR (XOR-d) games we study are a subclass of the well-known linear games. We introduce a ‘constraint graph’ associated to such a game, with the constraints defining the game represented by an edge-coloring of the graph. We use the graph-theoretic characterization to relate the task of finding equivalent games to the notion of signed graphs and switching equivalence from graph theory. We relate the problem of computing the classical value of single-party anti-correlation XOR games to finding the edge bipartization number of a graph, which is known to be MaxSNP hard, and connect the computation of the classical value of XOR-d games to the identification of specific cycles in the graph. We construct an orthogonality graph of the game from the constraint graph and study its Lovász theta number as a general upper bound on the quantum value even in the case of single-party contextual XOR-d games. XOR-d games possess appealing properties for use in device-independent applications such as randomness of the local correlated outcomes in the optimal quantum strategy. We study the possibility of obtaining quantum algebraic violation of these games, and show that no finite XOR-d game possesses the property of pseudo-telepathy leaving the frequently used chained Bell inequalities as the natural candidates for such applications. We also show this lack of pseudo-telepathy for multi-party XOR-type inequalities involving two-body correlation functions.

Generalized xor games withdoutcomes and the task of nonlocal computation

A natural generalization of the binary XOR games to the class of XOR-d games with $d > 2$ outcomes is studied. We propose an algebraic bound to the quantum value of these games and use it to derive several interesting properties of these games. As an example, we re-derive in a simple manner a recently discovered bound on the quantum value of the CHSH-d game for prime $d$. It is shown that no total function XOR-d game with uniform inputs can be a pseudo-telepathy game, there exists a quantum strategy to win the game only when there is a classical strategy also. We then study the principle of lack of quantum advantage in the distributed non-local computation of binary functions which is a well-known information-theoretic principle designed to pick out quantum correlations from amongst general no-signaling ones. We prove a large-alphabet generalization of this principle, showing that quantum theory provides no advantage in the task of non-local distributed computation of a restricted class of functions with $d$ outcomes for prime $d$, while general no-signaling boxes do. Finally, we consider the question whether there exist two-party tight Bell inequalities with no quantum advantage, and show that the binary non-local computation game inequalities for the restricted class of functions are not facet defining for any number of inputs.

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

We show that for any $\varepsilon>0$ the problem of finding a factor $(2-\varepsilon)$ 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$\neq$NP. They can be thought of as an extension of H\aastad'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 $53$rd FOCS, IEEE, Piscataway, NJ, 2012, pp. 243--252]. Our results demonstrate t...

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.