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.
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.
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.
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.
We study multiprover interactive proof systems. The power of classical multiprover interactive proof systems, in which the provers do not share entanglement, was characterized in a famous work by Babai, Fortnow, and Lund (Computational Complexity 1991), whose main result was the equality MIP = NEXP. The power of quantum multiprover interactive proof systems, in which the provers are allowed to share entanglement, has proven to be much more difficult to characterize. The best known lower-bound on MIP* is NEXP ⊆ MIP* due to Ito and Vidick (FOCS 2012). As for upper bounds, MIP* could be as large as RE, the class of recursively enumerable languages. The main result of this work is the inclusion NEEXP = NTIME[22poly(n)] ⊆ MIP*. This is an exponential improvement over the prior lower bound and shows that proof systems with entangled provers are at least exponentially more powerful than classical provers. In our protocol the verifier delegates a classical, exponentially large MIP protocol for NEEXP to two entangled provers: the provers obtain their exponentially large questions by measuring their shared state, and use a classical PCP to certify the correctness of their exponentially-long answers. For the soundness of our protocol, it is crucial that each player should not only sample its own question correctly but also avoid performing measurements that would reveal the other player's sampled question. We ensure this by commanding the players to perform a complementary measurement, relying on the Heisenberg uncertainty principle to prevent the forbidden measurements from being performed.
We study multiprover interactive proof systems. The power of classical multiprover interactive proof systems, in which the provers do not share entanglement, was characterized in a famous work by Babai, Fortnow, and Lund (Computational Complexity 1991), whose main result was the equality MIP = NEXP. The power of quantum multiprover interactive proof systems, in which the provers are allowed to share entanglement, has proven to be much more difficult to characterize. The best known lower-bound on MIP* is NEXP, due to Ito and Vidick (FOCS 2012). As for upper bounds, MIP* could be as large as RE, the class of recursively enumerable languages. The main result of this work is the inclusion of NEEXP (nondeterministic doubly exponential time) in MIP*. This is an exponential improvement over the prior lower bound and shows that proof systems with entangled provers are at least exponentially more powerful than classical provers. In our protocol the verifier delegates a classical, exponentially large MIP protocol for NEEXP to two entangled provers: the provers obtain their exponentially large questions by measuring their shared state, and use a classical PCP to certify the correctness of their exponentially-long answers. For the soundness of our protocol, it is crucial that each player should not only sample its own question correctly but also avoid performing measurements that would reveal the other player's sampled question. We ensure this by commanding the players to perform a complementary measurement, relying on the Heisenberg uncertainty principle to prevent the forbidden measurements from being performed.
The complexity of free games with two or more classical players was essentially settled by Aaronson, Impagliazzo, and Moshkovitz (CCC’14). In the quantum world, there are two complexity classes that can be considered quantum analogues of classical free games: (1) AM*, the multiprover interactive proof class corresponding to free games with entangled players, and, somewhat less obviously, (2) BellQMA(2), the class of quantum Merlin-Arthur proof systems with two unentangled Merlins, whose proof states are separately measured by Arthur. In this work, we make significant progress towards a tight characterization of both of these classes. (1) We show a BellQMA(2) protocol for 3SAT on n variables, where the total amount of communication is Õ(√n). This answers an open question of Chen and Drucker (2010) and also shows, conditional on ETH, that the algorithm of Brandão, Christandl and Yard (STOC’11) for optimizing over separable states is tight up to logarithmic factors. (2) We show that AM* with nprovers = 2, question length O(1), and answer-length log(n) is equal to RE, i.e. that free entangled games with constant-sized questions are as powerful as general entangled games. (In contrast, Aaronson, Impagliazzo and Moshkovitz show that classical free games are much weaker than general classical games.) We show this using a question “hyper-compression” theorem that iteratively applies the introspection technique of Ji et al. (2020). Our result is a significant improvement over the headline result of Ji et al., whose MIP* protocol for the halting problem has (n)-sized questions and answers. (3) By the same techniques, we obtain a zero-gap AM* protocol for a Π2 complete language with constant-size questions and almost logarithmically (O(logn · log* n)) large answers, improving on the headline result of Mousavi, Nezhadi and Yuen (STOC’22). (4) Using a connection to the nonuniform complexity of the halting problem we show that any MIP* protocol for RE requires Ω(logn) bits of communication. It follows that our results in item 3 are optimal up to an O(log* n) factor, and that the gapless compression theorems of Mousavi, Nezhadi and Yuen are asymptotically optimal. We conjecture that these bounds can be saturated in the gapped case as well.
We describe two generic ways to make multi-prover classical games resistant against entangled provers. The first uses quantum communication and a quantum verifier, the second adds an additional prover. This leads to several new results on the power of proof systems with entangled provers. We show that NEXP ⊆ QMIP1,s(2, 1) and NEXP ⊆ MIP1,s(3, 1) with soundness s = 1− 2− poly(n) and PSPACE ⊆ MIP1,s(2, 1) with soundness s = 1 − 1/ poly(n), providing the first non-trivial bounds in this setting. Moreover, our results imply that, unless P = NP, the value of entangled prover games cannot be computed by semi-definite programs that are polynomial in the size of the verifier’s system, a method that has been successful for more restricted quantum games. ∗also at CNRS & LRI, Univerite de Paris-Sud, Orsay, France †Partially supported by the European Commission under the Integrated Project Qubit Applications (QAP) funded by the IST directorate as Contract Number 015848 and by an Alon Fellowship of the Israeli Higher Council of Academic Research. ‡Part of this work was completed at Caltech. Supported by the National Science Foundation under Grants PHY-0456720 and CCF-0524828, by EU project QAP, and by NWO VICI project 639-023-302. Part of this research has been funded by the Dutch BSIK/BRICKS project. §Work partly done while at LRI, Univ. de Paris-Sud, Orsay.