{"id":{"repo_id":"mit","oai_identifier":"oai:dspace.mit.edu:1721.1/142811"},"canonical_url":"https://search.dev.ndltd.org/etd/mit/oai:dspace.mit.edu:1721.1/142811","repository":{"repo_id":"mit","name":"MIT","base_url":"https://dspace.mit.edu/oai/request"},"display":{"title":"Identifying Perfect Nonlocal Games","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_html":"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_has_math":false,"creators":["Bene Watts, Adam"],"institution":"Massachusetts Institute of Technology","degree_name":"Doctoral","degree_level":null,"degree_discipline":null,"degree_department":"Massachusetts Institute of Technology. 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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."]},{"key":"dc:description.degree","label":"Dc Description Degree","values":["Ph.D."]},{"key":"dc:title","label":"Title","values":["Identifying Perfect Nonlocal Games"]}]}],"canonical_facts":{"dc:contributor.advisor":["Harrow, Aram W."],"dc:contributor.department":["Massachusetts Institute of Technology. Department of Physics"],"dc:creator":["Bene Watts, Adam"],"dc:date.accessioned":["2022-05-31T13:29:44Z"],"dc:date.available":["2022-05-31T13:29:44Z"],"dc:date.issued":["2021-09"],"dc:description.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."],"dc:description.degree":["Ph.D."],"dc:identifier.uri":["https://hdl.handle.net/1721.1/142811"],"dc:publisher":["Massachusetts Institute of Technology"],"dc:rights":["In Copyright - Educational Use Permitted","Copyright MIT"],"dc:rights.uri":["http://rightsstatements.org/page/InC-EDU/1.0/"],"dc:title":["Identifying Perfect Nonlocal Games"],"dc:type":["Thesis"],"thesis:degree_name":["Doctoral","Doctor of Philosophy"]},"updated_at":"2026-07-22T22:20:49Z"}