• DOI:10.1109/CCC.2008.26
• Corpus ID: 16457957

The Quantum Moment Problem and Bounds on Entangled Multi-prover Games

@article{Doherty2008TheQM,
  title={The Quantum Moment Problem and Bounds on Entangled Multi-prover Games},
  author={Andrew C. Doherty and Yeong-Cherng Liang and Ben Toner and Stephanie Wehner},
  journal={2008 23rd Annual IEEE Conference on Computational Complexity},
  year={2008},
  pages={199-210}
}

• A. Doherty, Yeong-Cherng Liang, +1 author S. Wehner
• Published 31 March 2008
• Mathematics
• 2008 23rd Annual IEEE Conference on Computational Complexity

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.