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Harrow, Aram W. Bene Watts, Adam Massachusetts Institute of Technology. Department of Physics 2022-05-31T13:29:44Z 2022-05-31T13:29:44Z 2021-09 2022-05-25T19:52:53.238Z https://hdl.handle.net/1721.1/142811 This thesis is about nonlocal games. These “games” are really interactive tests in which a verifier checks the correlations that can be produced by non-communicating players. We study the class of commuting operator correlations: correlations which can by produced by players who make commuting measurements on some shared entangled state. This thesis contains following results: • A general algebraic characterization of games with a “perfect” commuting operator strategy, i.e. games with a winning correlation that can be produced exactly by commuting operator measurements. This characterization is built on a key result in non-commutative algebraic geometry known as a (non-commutative) Nullstellensatz. • A sufficient condition for a class of nonlocal games called XOR games to have a perfect commuting operator strategy. This condition can be checked in polynomial time, and can be understood either as non-existence of a combinatorial object called a PREF (the noPREF condition) or as non existence of a solution to an instance of the subgroup membership problem in a specially constructed group. • A family of simple one-qubit-per-player strategies we call MERP strategies, which we show are optimal for any XOR game which has a perfect commuting operator strategy by the noPREF condition. • Proofs that the noPREF condition is both necessary and sufficient for symmetric XOR games and 3 player XOR games. • Explicit constructions of several families of XOR games with interesting properties. • An analysis of randomly generated XOR games using the noPREF condition and the first moment method. Ph.D. Massachusetts Institute of Technology In Copyright - Educational Use Permitted Copyright MIT http://rightsstatements.org/page/InC-EDU/1.0/ Identifying Perfect Nonlocal Games Thesis application/pdf Doctoral Doctor of Philosophy Publication unknown unknown <Publication xmlns="https://www.openaire.eu/cerif-profile/1.1/" id="0a1ed1c4-1467-451f-afaa-751a03af8d49"> <Type xmlns="https://www.openaire.eu/cerif-profile/vocab/COAR_Publication_Types">http://purl.org/coar/resource_type/c_1843</Type> <Title>Identifying Perfect Nonlocal Games</Title> <PublishedIn> <Publication> </Publication> </PublishedIn> <PublicationDate>2021-09</PublicationDate> <Authors> <Author> <DisplayName>Bene Watts, Adam</DisplayName> <Affiliation> <OrgUnit> </OrgUnit> </Affiliation> </Author> </Authors> <Editors> </Editors> <Publishers> <Publisher> <DisplayName>Massachusetts Institute of Technology</DisplayName> <OrgUnit /> </Publisher> </Publishers> <License>http://rightsstatements.org/page/InC-EDU/1.0/</License> <Abstract>This thesis is about nonlocal games. These “games” are really interactive tests in which a verifier checks the correlations that can be produced by non-communicating players. We study the class of commuting operator correlations: correlations which can by produced by players who make commuting measurements on some shared entangled state. This thesis contains following results: • A general algebraic characterization of games with a “perfect” commuting operator strategy, i.e. games with a winning correlation that can be produced exactly by commuting operator measurements. This characterization is built on a key result in non-commutative algebraic geometry known as a (non-commutative) Nullstellensatz. • A sufficient condition for a class of nonlocal games called XOR games to have a perfect commuting operator strategy. This condition can be checked in polynomial time, and can be understood either as non-existence of a combinatorial object called a PREF (the noPREF condition) or as non existence of a solution to an instance of the subgroup membership problem in a specially constructed group. • A family of simple one-qubit-per-player strategies we call MERP strategies, which we show are optimal for any XOR game which has a perfect commuting operator strategy by the noPREF condition. • Proofs that the noPREF condition is both necessary and sufficient for symmetric XOR games and 3 player XOR games. • Explicit constructions of several families of XOR games with interesting properties. • An analysis of randomly generated XOR games using the noPREF condition and the first moment method.</Abstract> <Access xmlns="http://purl.org/coar/access_right" > </Access> </Publication>