The class $\MIP^*$ of promise problems that can be decided through an interactive proof system with multiple entangled provers provides a complexity-theoretic framework for the exploration of the nonlocal properties of entanglement. Little is known about the power of this class. The only proposed approach for establishing upper bounds is based on a hierarchy of semidefinite programs introduced independently by Pironio et al. and Doherty et al. This hierarchy converges to a value that is only known to coincide with the provers' maximum success probability in a given proof system under a plausible but difficult mathematical conjecture, Connes' embedding conjecture. No bounds on the rate of convergence are known. We introduce a rounding scheme for the hierarchy, establishing that any solution to its $N$-th level can be mapped to a strategy for the provers in which measurement operators associated with distinct provers have pairwise commutator bounded by $O(\ell^2/\sqrt{N})$ in operator norm, where $\ell$ is the number of possible answers per prover. Our rounding scheme motivates the introduction of a variant of $\MIP^*$, called $\MIP_δ^*$, in which the soundness property is required to hold as long as the commutator of operations performed by distinct provers has norm at most $δ$. Our rounding scheme implies the upper bound $\MIP_δ^* \subseteq \DTIME(\exp(\exp(\poly)/δ^2))$. In terms of lower bounds we establish that $\MIP^*_{2^{-\poly}}$, with completeness $1$ and soundness $1-2^{-\poly}$, contains $\NEXP$. The relationship of $\MIP_δ^*$ to $\MIPstar$ has connections with the mathematical literature on approximate commutation. Our rounding scheme gives an elementary proof that the Strong Kirchberg Conjecture implies that $\MIPstar$ is computable. We discuss applications to device-independent cryptography.
The class \MIP∗ of promise problems that can be decided through an interactive proof system with multiple entangled provers provides a complexity-theoretic framework for the exploration of the nonlocal properties of entanglement. Little is known about the power of this class. The only proposed approach for establishing upper bounds is based on a hierarchy of semidefinite programs introduced independently by Pironio et al. and Doherty et al. This hierarchy converges to a value that is only known to coincide with the provers' maximum success probability in a given proof system under a plausible but difficult mathematical conjecture, Connes' embedding conjecture. No bounds on the rate of convergence are known. We introduce a rounding scheme for the hierarchy, establishing that any solution to its N-th level can be mapped to a strategy for the provers in which measurement operators associated with distinct provers have pairwise commutator bounded by O(ℓ2/N) in operator norm, where ℓ is the number of possible answers per prover. Our rounding scheme motivates the introduction of a variant of \MIP∗, called \MIPδ∗, in which the soundness property is required to hold as long as the commutator of operations performed by distinct provers has norm at most δ. Our rounding scheme implies the upper bound \MIPδ∗⊆\DTIME(exp(exp(\poly)/δ2)). In terms of lower bounds we establish that \MIP2−\poly∗, with completeness 1 and soundness 1−2−\poly, contains \NEXP. The relationship of \MIPδ∗ to \MIPstar has connections with the mathematical literature on approximate commutation. Our rounding scheme gives an elementary proof that the Strong Kirchberg Conjecture implies that \MIPstar is computable. We discuss applications to device-independent cryptography.
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.
We study the problem of approximating the commuting-operator value of a two-player non-local game. It is well-known that it is NP-complete to decide whether the classical value of a non-local game is 1 or 1−ε. Furthermore, as long as ε is small enough, this result does not depend on the gap ε. In contrast, a recent result of Fitzsimons, Ji, Vidick, and Yuen shows that the complexity of computing the quantum value grows without bound as the gap ε decreases. In this paper, we show that this also holds for the commuting-operator value of a game. Specifically, in the language of multi-prover interactive proofs, we show that the power of MIPco(2,1,1,s) (proofs with two provers, one round, completeness probability 1, soundness probability s, and commuting-operator strategies) can increase without bound as the gap 1−s gets arbitrarily small. Our results also extend naturally in two ways, to perfect zero-knowledge protocols, and to lower bounds on the complexity of computing the approximately-commuting value of a game. Thus we get lower bounds on the complexity class PZK-MIPδco(2,1,1,s) of perfect zero-knowledge multi-prover proofs with approximately-commuting operator strategies, as the gap 1−s gets arbitrarily small. While we do not know any computable time upper bound on the class MIPco, a result of the first author and Vidick shows that for s=1−1/poly(f(n)) and δ=1/poly(f(n)), the class MIPδco(2,1,1,s), with constant communication from the provers, is contained in TIME(exp(poly(f(n)))). We give a lower bound of coNTIME(f(n)) (ignoring constants inside the function) for this class, which is tight up to polynomial factors assuming the exponential time hypothesis.
We study the problem of approximating the commuting-operator value of a two-player non-local game. It is well-known that it is NP-complete to decide whether the classical value of a non-local game is 1 or 1- epsilon, promised that one of the two is the case. Furthermore, as long as epsilon is small enough, this result does not depend on the gap epsilon. In contrast, a recent result of Fitzsimons, Ji, Vidick, and Yuen shows that the complexity of computing the quantum value grows without bound as the gap epsilon decreases. In this paper, we show that this also holds for the commuting-operator value of a game. Specifically, in the language of multi-prover interactive proofs, we show that the power of MIP^{co}(2,1,1,s) (proofs with two provers, one round, completeness probability 1, soundness probability s, and commuting-operator strategies) can increase without bound as the gap 1-s gets arbitrarily small. Our results also extend naturally in two ways, to perfect zero-knowledge protocols, and to lower bounds on the complexity of computing the approximately-commuting value of a game. Thus we get lower bounds on the complexity class PZK-MIP^{co}_{delta}(2,1,1,s) of perfect zero-knowledge multi-prover proofs with approximately-commuting operator strategies, as the gap 1-s gets arbitrarily small. While we do not know any computable time upper bound on the class MIP^{co}, a result of the first author and Vidick shows that for s = 1-1/poly(f(n)) and delta = 1/poly(f(n)), the class MIP^{co}_delta(2,1,1,s), with constant communication from the provers, is contained in TIME(exp(poly(f(n)))). We give a lower bound of coNTIME(f(n)) (ignoring constants inside the function) for this class, which is tight up to polynomial factors assuming the exponential time hypothesis.
It is known that there exist multi-prover interactive protocols (MIP protocols) for the complexity class NEXP, succinct MIP protocols for NP and multi-prover interactive protocols with shared entanglement (MIP∗ protocols) for RE. This extraordinary power of multi-prover interactive proof systems comes from the assumption that provers do not communicate with each other during the protocols. If they are allowed to communicate freely, the setting is the same as in the single-prover case, and the computational power of the system becomes significantly weaker.
In this paper, we investigate for the first time the setting where communication (i.e., leakage of information) between provers is allowed but bounded. We introduce two techniques to approach this question and show that multi-prover interactive proof systems are robust against some amount of leakage. Our first technique is based on parallel repetition theorems. We apply it to show that for any polynomial p, we can construct two-prover one-round MIP and MIP∗ protocols for NEXP and RE, respectively, that are robust against p(n) bits of leakage. We further derive our second technique to convert any low-soundness PCP construction to a two-prover one-round MIP protocol for NP robust against leakage. We also discuss the relation between robustness against leakage in multi-prover interactive proof systems and the Sliding Scale Conjecture in the PCP literature.
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.