The notions of entanglement and nonlocality are among the most striking ingredients found in quantum information theory. One tool to better understand these notions is the model of nonlocal games; a mathematical framework that abstractly models a physical system. The simplest instance of a nonlocal game involves two players, Alice and Bob, who are not allowed to communicate with each other once the game has started and who play cooperatively against an adversary referred to as the referee. The focus of this thesis is a class of games called extended nonlocal games, of which nonlocal games are a subset. In an extended nonlocal game, the players initially share a tripartite state with the referee. In such games, the winning conditions for Alice and Bob may depend on outcomes of measurements made by the referee, on its part of the shared quantum state, in addition to Alice and Bob's answers to the questions sent by the referee. We build up the framework for extended nonlocal games and study their properties and how they relate to nonlocal games.
The notions of entanglement and nonlocality are among the most striking ingredients found in quantum information theory. One tool to better understand these notions is the model of nonlocal games; a mathematical framework that abstractly models a physical system. The simplest instance of a nonlocal game involves two players, Alice and Bob, who are not allowed to communicate with each other once the game has started and who play cooperatively against an adversary referred to as the referee. The focus of this thesis is a class of games called extended nonlocal games, of which nonlocal games are a subset. In an extended nonlocal game, the players initially share a tripartite state with the referee. In such games, the winning conditions for Alice and Bob may depend on outcomes of measurements made by the referee, on its part of the shared quantum state, in addition to Alice and Bob's answers to the questions sent by the referee. We build up the framework for extended nonlocal games and study their properties and how they relate to nonlocal games.
We study a generalization of non-local games—which we call extended non-local games —in which the players, Alice and Bob, initially share a tripartite quantum state with the referee. In such games, the winning conditions for Alice and Bob may depend on the outcomes of measurements made by the referee, on its part of the shared quantum state, in addition to Alice and Bob's answers to randomly selected questions. Our study of this class of games was inspired by the monogamy-of-entanglement games introduced by Tomamichel, Fehr, Kaniewski and Wehner, which they also generalize. We prove that a natural extension of the Navascués–Pironio–Acín hierarchy of semidefinite programmes converges to the optimal commuting measurement value of extended non-local games, and we prove two extensions of results of Tomamichel et al. concerning monogamy-of-entanglement games.
Non-local games are studied in quantum information because they provide a simple way for proving the difference between the classical world and the quantum world. A non-local game is a cooperative game played by 2 or more players against a referee. The players cannot communicate but may share common random bits or a common quantum state. A referee sends an input xi to the ith player who then responds by sending an answer ai to the referee. The players win if the answers ai satisfy a condition that may depend on the inputs xi. Typically, non-local games are studied in a framework where the referee picks the inputs from a known probability distribution. We initiate the study of non-local games in a worst-case scenario when the referee's probability distribution is unknown and study several non-local games in this scenario.
Abstract This paper considers the decidability of fully quantum nonlocal games with noisy maximally entangled states. Fully quantum nonlocal games are a generalization of nonlocal games, where both questions and answers are quantum and the referee performs a binary POVM measurement to decide whether they win the game after receiving the quantum answers from the players. The quantum value of a fully quantum nonlocal game is the supremum of the probability that they win the game, where the supremum is taken over all the possible entangled states shared between the players and all the valid quantum operations performed by the players. The seminal work MIP∗=RE~\cite{JNVWY'20,JNVWYuen'20} implies that it is undecidable to approximate the quantum value of a fully nonlocal game. This still holds even if the players are only allowed to share (arbitrarily many copies of) maximally entangled states. This paper investigates the case that the shared maximally entangled states are noisy. We prove that there is a computable upper bound on the copies of noisy maximally entangled states for the players to win a fully quantum nonlocal game with a probability arbitrarily close to the quantum value. This implies that it is decidable to approximate the quantum values of these games. Hence, the hardness of approximating the quantum value of a fully quantum nonlocal game is not robust against the noise in the shared states. This paper is built on the framework for the decidability of non-interactive simulations of joint distributions~\cite{7782969,doi:10.1137/1.9781611975031.174,Ghazi:2018:DRP:3235586.3235614} and generalizes the analogous result for nonlocal games in~\cite{qin2021nonlocal}. We extend the theory of Fourier analysis to the space of super-operators and prove several key results including an invariance principle and a dimension reduction for super-operators. These results are interesting in their own right and are believed to have further applications.
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.
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.
One of the major concepts introduced by quantum theory is the compatibility of quantum measurements. There are certain types of measurements that cannot be made at the same time. Thus,we say that measurements are compatible if they can be measured at the same time and others are incompatible. The other major concept of quantum mechanics is that of nonlocality which is one of the most counterintuitive concepts of quantum physics. This major concept is due to John Bell who showed that quantum mechanics is intrinsically non-local. Thus, we speak of violation of Bell's inequalities by quantum mechanics. Today, nonlocality is understood through nonlocal games. A nonlocal game consists of two or more players Alice and Bob playing against a referee. The referee will ask a number of questions to the players who will have to generate a number of answers using a classical or quantum strategy. It turns out that the maximum number of answers that Alice and Bob can generate is intrinsically linked to a tensor norm characterizing the game. In this formalism, the use of classical strategies is related to the norm of the matrix of the game itself, so the violation of Bell's inequalities results in a strict inequality between the tensor norms. The aim of this thesis is to understand the incompatibility of quantum measures and the link with Bell's inequalities. First, we introduced the compatibility of quantum measures from a new point of view, and analyzed the types of noise that can be made to make the system compatible. This new point of view consists in understanding and analyzing the effect of the dimension of the Hilbert space on the incompatibility of measurements. Moreover, in order to make the measurements compatible, we can introduce the effect of a noise. As an application, some states known as MUB are incompatible in nature, we show that even if we add noise to the MUB it remains incompatible, there is an isometry and a Hilbert space of smaller dimension making the MUB compatible. In a second step, we have analyzed the intrinsic link between the incompatibility of quantum measurements and the violation of Bell's inequalities. For this purpose, we considered the framework of non-local games, where Alice's measurements are fixed. It is known that a violation of Bell's inequalities requires the use of incompatible measurements. On the other hand, if Alice wants to know if she will observe a violation of Bell's inequalities if she uses incompatible measures. To do this, she must compute two tensor norms of a tensor constructed from her measurements. These tensorial norms will characterize on the one hand the compatibility of Alice's measurements and on the other hand the violation of Bell's inequalities. In this natural framework, to understand the link between the incompatibility of quantum measurements and the violation of Bell's inequalities, we have to compare the two tensorial norms. Now, it turns out that for the CHSH game these two norms are equal, but it can be shown generally that they are not. We can ask ourselves if there are other types of games satisfying this equality of the tensorial norms? It turns out that we have shown that with sufficient conditions, only the CHSH game with a multiplicative constant gives the equality between the tensor norms.