The quantum moment problem and bounds on entangled multi-prover games

Andrew C. Doherty, Ben Toner, Yeong Cherng Liang, Stephanie Wehner

Research output: Chapter in Book/Report/Conference proceedingConference contribution

105 Citations (Scopus)

Abstract

We study the quantum moment problem: Given a conditional probability distribution together with some polynomial constraints, does there exist a quantum state p and a collection of measurement operators such that (i) the probability of obtaining a particular outcome when a particular measurement is performed on p 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 Navascués, Pironio and Acín /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.

Original languageEnglish
Title of host publicationProceedings - 23rd Annual IEEE Conference on Computational Complexity, CCC 2008
Pages199-210
Number of pages12
DOIs
Publication statusPublished - 2008
Event23rd Annual IEEE Conference on Computational Complexity, CCC 2008 - College Park, MD, United States
Duration: 2008 Jun 232008 Jun 26

Publication series

NameProceedings of the Annual IEEE Conference on Computational Complexity
ISSN (Print)1093-0159

Other

Other23rd Annual IEEE Conference on Computational Complexity, CCC 2008
Country/TerritoryUnited States
CityCollege Park, MD
Period08-06-2308-06-26

All Science Journal Classification (ASJC) codes

  • Software
  • Theoretical Computer Science
  • Computational Mathematics

Fingerprint

Dive into the research topics of 'The quantum moment problem and bounds on entangled multi-prover games'. Together they form a unique fingerprint.

Cite this