We establish the first hardness results for the problem of computing the value of one-round games played by a verifier and a team of provers who can share quantum entanglement. In particular, we show that it is NP-hard to approximate within an inverse polynomial the value of a one-round game with (i) a quantum verifier and two entangled provers or (ii) a classical verifier and three entangled provers. Previously it was not even known if computing the value exactly is NP-hard. We also describe a mathematical conjecture, which, if true, would imply hardness of approximation of entangled-prover games to within a constant. Using our techniques we also show that every language in PSPACE has a two-prover one-round interactive proof system with perfect completeness and soundness $1-1/\,\mathrm{poly}$ even against entangled provers. We start our proof by describing two ways to modify classical multiprover games to make them resistant to entangled provers. We then show that a strategy for the modified game that uses entanglement can be “rounded” to one that does not. The results then follow from classical inapproximability bounds. Our work implies that, unless $\mathrm{P}=\mathrm{NP}$, the values of entangled-prover games cannot be computed by semidefinite programs that are polynomial in the size of the verifier's system, a method that has been successful for more restricted quantum games.

We consider one-round games between a classical verifier and two provers who share entanglement. We show that when the constraints enforced by the verifier are “unique” constraints (i.e., permutations), the value of the game can be well approximated by a semidefinite program (SDP). Essentially the only algorithm known previously was for the special case of binary answers, as follows from the work of Tsirelson in 1980. Among other things, our result implies that the variant of the unique games conjecture where we allow the provers to share entanglement is false. Our proof is based on a novel “quantum rounding technique,” showing how to take a solution to an SDP and transform it into a strategy for entangled provers. Using our approximation by an SDP, we also show a parallel repetition theorem for unique entangled games.

Assume Alice and Bob share some bipartite d-dimensional quantum state. A well-known result in quantum mechanics says that by performing two-outcome measurements, Alice and Bob can produce correlations that cannot be obtained locally, i.e., with shared randomness alone. We show that by using only two bits of communication, Alice and Bob can classically simulate any such correlations. All previous protocols for exact simulation required the communication to grow to infinity with the dimension d. Our protocol and analysis are based on a power series method, resembling Krivine's bound on Grothendieck's constant, and on the computation of volumes of spherical tetrahedra.

We study the quantum moment problem: given a conditional probability distribution together with some polynomial constraints, does there exist a quantum state rho and a collection of measurement operators such that (i) the probability of obtaining a particular outcome when a particular measurement is performed on rho is specified by the conditional probability distribution, and (ii) the measurement operators satisfy the constraints. For example, the constraints might specify that some measurement operators must commute. We show that if an instance of the quantum moment problem is unsatisfiable, then there exists a certificate of a particular form proving this. Our proof is based on a recent result in algebraic geometry, the noncommutative Positivstellensatz of Helton and McCullough [Trans. Amer. Math. Soc., 356(9):3721, 2004]. A special case of the quantum moment problem is to compute the value of one-round multi-prover games with entangled provers. Under the conjecture that the provers need only share states in finite-dimensional Hilbert spaces, we prove that a hierarchy of semidefinite programs similar to the one given by Navascues, Pironioand Acin [Phys. Rev. Lett., 98:010401, 2007] converges to the entangled value of the game. Under this conjecture, it would follow that the languages recognized by a multi-prover interactive proof system where the provers share entanglement are recursive.

We establish the first hardness results for the problem of computing the value of one-round games played by a referee and a team of players who can share quantum entanglement. In particular, we show that it is NP-hard to approximate within an inverse polynomial the value of a one-round game with (i) quantum referee and two entangled players or (ii) classical referee and three entangled players. Previously it was not even known if computing the value exactly is NP-hard. We also describe a mathematical conjecture, which, if true, would imply hardness of approximation to within a constant.We start our proof by describing two ways to modify classical multi-player games to make them resistant to entangled players. We then show that a strategy for the modified game that uses entanglement can be "rounded'' to one that does not. The results then follow from classical inapproximability bounds. Our work implies that, unless P = NP, the values of entangled-player games cannot be computed by semidefinite programs that are polynomial in the size of the referee's system, a method that has been successful for more restricted quantum games.

We consider one-round games between a classical verifier and two provers who share entanglement. We show that when the constraints enforced by the verifier are `unique' constraints (i.e., permutations), the value of the game can be well approximated by a semidefinite program. Essentially the only algorithm known previously was for the special case of binary answers, as follows from the work of Tsirelson in 1980. Among other things, our result implies that the variant of the unique games conjecture where we allow the provers to share entanglement is false. Our proof is based on a novel `quantum rounding technique', showing how to take a solution to an SDP and transform it to a strategy for entangled provers. Using our approximation by a semidefinite program we also show a parallel repetition theorem for unique entangled games.

We investigate the existence of secure bit commitment protocols in the convex framework for probabilistic theories. The theory makes only minimal assumptions, and can be used to formalize quantum theory, classical probability theory, and a host of other possibilities. We prove that in all such theories that are locally non-classical but do not have entanglement, there exists a bit commitment protocol that is exponentially secure in the number of systems used.

Assume Alice and Bob share some bipartite d-dimensional quantum state. As is well known, by performing two-outcome measurements, Alice and Bob can produce correlations that cannot be obtained classically. We show that by using only two bits of communication, Alice and Bob can classically simulate any such correlations. All previous protocols for exact simulation required the communication to grow to infinity with the dimension d. Our protocol and analysis are based on a power series method, resembling Krivine's bound on Grothendieck's constant, and on the computation of volumes of spherical tetrahedra.