Identifying Perfect Nonlocal Games
Name
benewatts-abenewat-phd-physics-2021-thesis.pdf
Description
Thesis PDF
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1.39 MB
Format
Adobe PDF
Checksum (MD5)
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Author(s)
Bene Watts, Adam
Advisor(s)
Harrow, Aram W.
Date Issued
September 2021
Publisher
Massachusetts Institute of Technology
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.
MIT Department
Massachusetts Institute of Technology. Department of Physics
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