Quantum entanglement, and the resulting peculiar non-classical correlations are one of the most counter-intuitive aspects of quantum mechanics.
The formalism of interactive games from computational complexity theory provides a useful framework in which to understand the power of entanglement. In an interactive game, a verifier interacts with a number of infinitely powerful provers who are allowed to share quantum entanglement but otherwise can't communicate. The ability of the verifier to extract useful information from the provers, whom he does not trust, provides an interesting measure of the ability of the provers to coordinate using their shared entanglement. Two particular interactive games we'll look at are Magic Square and 3SAT.
The Magic Square game is the iconic example of a game where two classical provers cannot perfectly coordinate their strategies using shared randomness, but quantum provers with shared entanglement can win with probability 1. We show that by adding an extra prover, we disallow perfect cheating. For 3SAT with three provers we also show that perfect cheating is not possible.
We then generalize the results for Magic Square and 3SAT by looking at non-commuting provers, a superset of entangled provers (communication is allowed, but operators applied by different provers must commute). Using this method, we obtain a generalized Tsirelson inequality that we apply to the Magic Square. Hence, we are able to give provably optimal strategies for the general Magic Square with n players. We also recover a similar result for 3SAT as with entangled provers, and we improve on it by showing that the gap is inverse exponential in the input size.
The no-signaling principle, which forbids faster than light communication, is a fundamental constraint on the non-local correlations resulting from quantum entanglement. However, no-signaling allows distributions that are more general than those arising from quantum mechanics. In particular, using a specific unit of general non-local correlation (the Popescu-Rohlich box), we show that there are classes that are equal to NEXP classically, but collapse to AM once such correlations are allowed. We also show that MIP where the verifier only looks at the XOR of the answers collapses to PSPACE. For the second approach, we show that by writing general non-local correlations as linear constraints, MIP is included in EXP under such correlations (vs NEXP classically).
We can extend these results to entangled quantum provers, by formulating an artificial MIP-like class built on a promise problem, that classically is equal to NEXP, but that also collapses to AM when quantum correlations are present.
Quantum entanglement, and the resulting peculiar non-classical correlations are one of the most counter-intuitive aspects of quantum mechanics.
The formalism of interactive games from computational complexity theory provides a useful framework in which to understand the power of entanglement. In an interactive game, a verifier interacts with a number of infinitely powerful provers who are allowed to share quantum entanglement but otherwise can't communicate. The ability of the verifier to extract useful information from the provers, whom he does not trust, provides an interesting measure of the ability of the provers to coordinate using their shared entanglement. Two particular interactive games we'll look at are Magic Square and 3SAT.
The Magic Square game is the iconic example of a game where two classical provers cannot perfectly coordinate their strategies using shared randomness, but quantum provers with shared entanglement can win with probability 1. We show that by adding an extra prover, we disallow perfect cheating. For 3SAT with three provers we also show that perfect cheating is not possible.
We then generalize the results for Magic Square and 3SAT by looking at non-commuting provers, a superset of entangled provers (communication is allowed, but operators applied by different provers must commute). Using this method, we obtain a generalized Tsirelson inequality that we apply to the Magic Square. Hence, we are able to give provably optimal strategies for the general Magic Square with n players. We also recover a similar result for 3SAT as with entangled provers, and we improve on it by showing that the gap is inverse exponential in the input size.
The no-signaling principle, which forbids faster than light communication, is a fundamental constraint on the non-local correlations resulting from quantum entanglement. However, no-signaling allows distributions that are more general than those arising from quantum mechanics. In particular, using a specific unit of general non-local correlation (the Popescu-Rohlich box), we show that there are classes that are equal to NEXP classically, but collapse to AM once such correlations are allowed. We also show that MIP where the verifier only looks at the XOR of the answers collapses to PSPACE. For the second approach, we show that by writing general non-local correlations as linear constraints, MIP is included in EXP under such correlations (vs NEXP classically).
We can extend these results to entangled quantum provers, by formulating an artificial MIP-like class built on a promise problem, that classically is equal to NEXP, but that also collapses to AM when quantum correlations are present.
Entanglement is at the heart of quantum mechanics. The nonlocal correlations that can be obtained from space-time separated measurements on an entangled state are a central feature which provably distinguish it from local theories. This dissertation studies entanglement through a computational viewpoint. We develop new insights into the complex nature of entanglement by studying its role in multiplayer games, in which cooperating, but non-communicating, players interact with a referee in an attempt to win a pre-specified game. On the one hand, the nonlocal correlations that entanglement allows may enable players using it to develop new colluding strategies, defeating previously secure protocols. On the other, the richness of this new resource may also be exploited in order to design new protocols, providing solutions to problems previously deemed impossible. We explore both aspects of this dual nature of entanglement, putting limits on its strength while at the same time showing how it can be put to profit to solve new computational problems. A major unresolved question on the computational complexity of multiplayer entangled games is the power of MIP*, the class of languages having entangled multi-prover interactive proofs: how does it relate to its purely classical analogue MIP, which was completely characterized through the fundamental equation MIP=NEXP? Since the players may use entanglement to increase their odds at colluding against the verifier, MIP* could potentially be a much weaker class than MIP. Indeed, for a long time it has been an open question whether two entangled provers are more useful than a single prover. In this thesis we resolve this question by showing that the class of languages having multiprover interactive proofs with entangled provers is at least as large as its classical counterpart: NEXP is included in MIP*. At the heart of this result is an analysis of the multilinearity test of Babai, Fortnow, and Lund in the presence of entanglement. The fact that this test remains sound gives a systematic way for a verifier to impose strong limits on the ability of entangled provers to collude against the verifier.Gap amplification is a fundamental primitive in the study of classical multiplayer games. While sequential repetition of a game always decreases the prover's maximum success probability at an exponential rate, the fact that parallel repetition also achieves a gap amplification is a highly non-trivial fact. We show that gap amplification can be performed in parallel even in the presence of entanglement between the provers. We adapt a technique which was originally introduced by Feige and Kilian and results in a polynomial rate of amplification.The phenomenon of monogamy of entanglement states, in first approximation, that if two parties are maximally entangled then they cannot simultaneously be entangled with a third party. We use this phenomenon in two distinct results. In the first, we show that the bits generated in our randomness-expansion protocol are certifiably random even from the point of view of a quantum adversary who may share prior entanglement with the provers. In addition, we prove the security against quantum adversaries of a randomness-efficient extractor construction originally due to Trevisan. This lets us transform the high-entropy bits that are generated in our protocol into ones that are almost indistinguishable from uniform by any adversary. More generally, we show how the monogamy of entanglement can be exploited to design multi-prover interactive proof systems that are partially entanglement-resistant. Quantitative bounds on the monogamy of entanglement have generally been elusive, and the analysis of our protocol demonstrates such a bound in a new context. The nonlocal correlations that can be created by entangled players provide a statistical means of differentiating them from classical, unentangled players. This is the main idea behind Bell inequalities, the violation of which demonstrates the nonlocality of quantum mechanics. We show how this phenomenon may be exploited to design a protocol in which the bits produced by successful players necessarily contain a large quantity of fresh randomness. The presence of randomness is guaranteed irrespective of the provers' actual strategy, as long as the sole constraint of no signaling is respected. Hence a statistical certification for the presence of randomness, a feat easily seen to be impossible to achieve classically. In order to manipulate the random bits produced in our protocol, and make them useful in cryptography, we give the first proof of security of a poly-logarithmic seed extractor secure against quantum adversaries. To achieve this we adapt the reconstruction paradigm originally introduced by Trevisan to the quantum setting. We study other ways in which entanglement may be used in interactive proof systems by also allowing a quantum interaction between the referee and the players. We show that, using entanglement, the class of QMIP* proof systems can be parallelized to only three rounds of interaction, and made public-coin, a property that does not hold in the absence of entanglement between the provers.
Nonlocal games provide a unified framework for studying the distinction between classical, quantum, and more general no-signaling correlations. In this work, we develop this perspective by connecting the Bell-locality framework to several complementary mathematical representations of nonlocal games and quantum strategies. We begin with local hidden-variable models, the CHSH inequality, and the role of Bell nonlocality as a device-independent witness of entanglement, and then introduce nonlocal games through the standard predicate/verifier formalism.
We next examine a set of representative examples, including XOR games, the GHZ game, graph-based coloring games, the Mermin-Peres magic square game, and Hardy's paradox as a related logical manifestation of nonlocality. Building on this foundation, we compare four closely related representation frameworks: conditional-probability and correlation descriptions, Bell-functional formulations, entangled-value optimization, and the quantum-operator approach together with the Navascues-Pironio-Acin (NPA) hierarchy. These viewpoints are then instantiated for the CHSH, magic square, and GHZ games, showing how each representation emphasizes a different aspect of the same underlying task.
Taken together, these examples show that nonlocal games can be studied simultaneously as geometric objects in correlation space, optimization problems over entangled resources, and operator-theoretic constructions. This multi-representation viewpoint clarifies the relation between Bell inequality violations, perfect quantum strategies, pseudo-telepathy, and semidefinite relaxations of quantum correlations.
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. Obviously, these commuting-operator provers are at least as powerful as usual isolated but prior-entangled provers, and thus, limits in the model with commuting-operator provers immediately give limits in the usual model with prior-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 three-prover one-round binary interactive proof system 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 three-prover one-round binary-answer game is equal to one or at most 1-1/p(n) for some polynomial p, where n is the number of questions. This is in contrast to the two-prover one-round binary-answer case, where the corresponding problem is efficiently decidable. Alternatively, NEXP has a three-prover one-round binary interactive proof system with perfect completeness and soundness 1 middot 2-poly.
Nonlocal games provide a unified framework for studying the distinction between classical, quantum, and more general no-signaling correlations. This review connects the Bell-locality framework to several complementary mathematical representations of nonlocal games and quantum strategies. We begin with local hidden-variable models, the CHSH inequality, and the role of Bell nonlocality as a device-independent witness of entanglement. We then introduce nonlocal games through the standard predicate/verifier formalism and discuss representative examples, including XOR games, the GHZgame, graph-based colouring games, the Mermin–Peres magic square game, and Hardy’s paradox as a related logical manifestation of nonlocality. The central contribution of the article is a comparative organisation of four closely related representation frameworks: conditional-probability andcorrelation descriptions, Bell-functional formulations, entangled-value optimisation, and the quantum-operator approach together with the Navascu´es–Pironio–Ac´ın hierarchy. These viewpoints are instantiated for the CHSH, magic square, and GHZ games, showing how each representation emphasises a different aspect of the same underlying task. The resulting synthesis clarifies the relation between Bell inequality violations, perfect quantum strategies, pseudo-telepathy, and semidefinite relaxations of quantum correlations.
Quantum de Finetti theorems are a useful tool in the study of correlations in quantum multipartite states. In this paper we prove two new quantum de Finetti theorems, both showing that under tests formed by local measurements in each of the subsystems one can get a much improved error dependence on the dimension of the subsystems. We also obtain similar results for non-signaling probability distributions. We give the following applications of the results to quantum complexity theory, polynomial optimization, and quantum information theory: We prove the optimality of the Chen-Drucker protocol for 3-SAT, under the assumption there is no subexponential-time algorithm for SAT. In the protocol a prover sends to a verifier √n polylog(n) unentangled quantum states, each composed of O(log(n)) qubits, as a proof of the satisfiability of a 3-SAT instance with n variables and O(n) clauses. The quantum verifier checks the validity of the proof by performing local measurements on each of the proofs and classically processing the outcomes. We show that any similar protocol with O(n1/2 - ε) qubits would imply a exp (n1 - 2ε polylog(n))-time algorithm for 3-SAT. We show that the maximum winning probability of free games (in which the questions to each prover are chosen independently) can be estimated by linear programming in time exp(O(log|Q| + log2|A|/ε2) ), with |Q| and |A| the question and answer alphabet sizes, respectively, matching the performance of a previously known algorithm due to Aaronson, Impagliazzo, Moshkovitz, and Shor. This result follows from a new monogamy relation for non-locality, showing that k-extendible non-signaling distributions give at most a O(k-1/2) advantage over classical strategies for free games. We also show that 3-SAT with n variables can be reduced to obtaining a constant error approximation of the maximum winning probability under entangled strategies of O(√n)-player one-round non-local games, in which only two players are selected to send O(√n)-bit messages. We show that the optimization of certain polynomials over the complex hypersphere can be performed in quasipolynomial time in the number of variables n by considering O(log(n)) rounds of the Sum-of-Squares (Parrilo/Lasserre) hierarchy of semidefinite programs. This can be considered an analogue to the hypersphere of a similar known results for the simplex. As an application to entanglement theory, we find a quasipolynomial-time algorithm for deciding multipartite separability. We consider a quantum tomography result due to Aaronson -- showing that given an unknown n-qubit state one can perform tomography that works well for most observables by measuring only O(n) independent and identically distributed (i.i.d.) copies of the state -- and relax the assumption of having i.i.d copies of the state to merely the ability to select subsystems at random from a quantum multipartite state. The proofs of the new quantum de Finetti theorems are based on information theory, in particular on the chain rule of mutual information. The results constitute improvements and generalizations of a recent de Finetti theorem due to Brandao, Christandl and Yard.
We prove a strong limitation on the ability of entangled provers to collude in a multiplayer game. Our main result is the first nontrivial lower bound on the class MIP* of languages having multi-prover interactive proofs with entangled provers, namely MIP* contains NEXP, the class of languages decidable in non-deterministic exponential time. While Babai, Fort now, and Lund (Computational Complexity 1991) proved the celebrated equality MIP = NEXP in the absence of entanglement, ever since the introduction of the class MIP* it was open whether shared entanglement between the provers could weaken or strengthen the computational power of multi-prover interactive proofs. Our result shows that it does not weaken their computational power: MIP* contains MIP. At the heart of our result is a proof that Babai, Fort now, and Lund's multilinearity test is sound even in the presence of entanglement between the provers, and our analysis of this test could be of independent interest. As a byproduct we show that the correlations produced by any entangled strategy which succeeds in the multilinearity test with high probability can always be closely approximated using shared randomness alone.