A Multi-prover Interactive Proof for NEXP Sound against Entangled Provers

Tsuyoshi Ito, Thomas Vidick
2012 2012 IEEE 53rd Annual Symposium on Foundations of Computer Science  
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, Fortnow, and Lund (Computational Complexity 1991) proved the celebrated equality MIP = NEXP in the absence of entanglement, ever since the
more » ... troduction 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 ⊆ MIP * . At the heart of our result is a proof that Babai, Fortnow, 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. 1 This was later improved [Weh06] to the inclusion of ⊕MIP * in the class of two-message single-prover interactive proofs QIP(2) ⊆ PSPACE [JUW09]. 2 It was recently shown that quantum messages are no more powerful than classical messages in single-prover interactive proof systems [JJUW11]: QIP = PSPACE. That result, however, has no direct relationship with our work: in our setting the messages remain classical; rather the "quantumness" manifests itself in the presence of entanglement between the provers, which is a notion that only arises when more than one prover is present. 3 A collection of distributions on the provers' answers, one for every tuple of questions, is no-signaling if, for any such distribution, its marginal on any subset of the provers is independent of the questions to the remaining provers. 4 This formulation was first observed by Daniel Preda. 2 The fact that entanglement, as a shared resource, is poorly understood is also reflected in the complete absence of reasonable upper bounds on the complexity class MIP * : while the inclusion MIP ⊆ NEXP is straightforward, we do not know of any limits on the dimension of entanglement that may be useful to the provers in a given interactive proof system, and as a result their maximum success probability is not even known to be computable (see [SW08, DLTW08, NPA08] for more on this aspect). Since existing protocols may no longer be sound in the presence of entanglement between the provers, previous work has focused on finding ways to modify a given protocol in a way that would make it entanglement resistant; that is, honest provers (in the case of a YES-instance) can convince the verifier without shared entanglement while dishonest provers (in the case of a NOinstance) cannot convince the verifier with high probability even with shared entanglement. This was the route taken in [KKMTV11, IKPSY08, IKM09], which introduced techniques to limit the provers' use of their entanglement. They proved non-trivial lower bounds on variants of the class MIP * , but with error bounds that are weaker than the standard definitions allow for. These relatively weak bounds came as a result of the "rounding" technique developed in these works: by adding additional constraints to the protocol, one ensures that optimal entangled strategies are in a sense close to classical, un-entangled strategies. This closeness, however, was shown using a rounding procedure that had a certain "local" flavor, inducing a large loss in the quality of the approximation. 5 In addition, [IKM09], based on [KKMTV11], showed that PSPACE has two-prover one-round interactive proofs with entangled provers, with perfect completeness and exponentially small soundness error. Prior to our work, this was the best lower bound known on single-round multiprover interactive proof systems with entanglement. Additional related work. Given the apparent difficulty of proving good lower bounds on the power of multi-prover interactive proof systems with entangled provers, researchers have studied a variety of related models. Maybe the most natural extension of MIP * consists in giving the verifier more power by allowing him to run in quantum polynomial-time, and exchange quantum messages with the provers. The resulting class is called QMIP * (the Q stands for "quantum verifier", while the * stands for "entangled provers"), and it was formally introduced in [KM03], where it was shown that QMIP * contains MIP * (indeed, the verifier can always force classical communication by systematically measuring the provers' answers in the computational basis). Recently Reichardt et al. [RUV12] showed that QMIP * = MIP * (the possibility of which had been suggested earlier in [BFK10]). Ben-Or et al. [BHP08] introduced a model in which the verifier is quantum and the provers are allowed communication but no entanglement, and showed that the resulting class contains NEXP. Other works attempt to characterize the power of MIP * systems using tensor norms [RT07, JPPVW10]; so far however such norms have either led to computable, but very imprecise, approximations, or have remained (to the best of our knowledge) intractable. Results Let MIP * (k, m, c, s) be the class of languages that can be decided by an m-round interactive proof system with k (possibly entangled) provers and with completeness c and soundness error s. 6 Our main result is the following. 5 See the "almost-commuting implies nearly-commuting" conjecture in [KKMTV11] for more on this aspect. 6 We refer to Section 2.2 for a more complete definition of the class MIP * .
doi:10.1109/focs.2012.11 dblp:conf/focs/ItoV12 fatcat:njeobvhslfaopkver2ul3u2dk4