Friday, January 22, 2021
4:00 p.m.

PhD Candidate:  Arthur Mehta
Supervisor:   Henry Yuen
Thesis title:  Entanglement and non-locality in games and graphs

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This thesis is primarily based on two collaborative works written by the author and several coauthors. These works are presented in Chapters 4 and 5 and are on the topics of quantum graphs, and self-testing via non-local games, respectively.

Quantum graph theory, also known as non-commutative graph theory, is an operator space generalization of graph theory. The independence number, and Lova’sz theta function were generalized to this setting by Duan, Severini, and Winter and two different version of the chromatic number were introduced by Stahlke and Paulsen. In Chapter 4, we introduce two new generalizations of the chromatic number to non-commutative graphs and provide an upper bound on the parameter of Stahlke. We provide a generalization of the graph complement and show the chromatic number of the orthogonal complement of a non-commutative graph is bounded below by its theta number. We also provide a generalization of both Sabidussi’s Theorem and Hedetniemi’s conjecture to non-commutative graphs.

The study of non-local games considers scenarios in which separated players collaborate to provide satisfying responses to questions given by a referee. The condition of separating players makes non-local games an excellent setting to gain insight into quantum phenomena such as entanglement and non-locality. Non-local games can also provide protocols known as self-tests. Self-testing allows an experimenter to interact classically with a black box quantum system and certify that a specific entangled state was present, and a specific set of measurements were performed. The most studied self-test is the CHSH game which certifies the presence of a single EPR entangled state and the use of anti-commuting Pauli measurements. In Chapter 5, we introduce an algebraic generalization of CHSH and obtain a self-test for non-Pauli operators resolving an open question posed by Coladangelo and Stark (QIP 2017). Our games also provide a self-test for states other than the maximally entangled state, and hence resolves the open question posed by Cleve and Mittal (ICALP 2012).

The results of Chapter 5 make use of sums of squares techniques in the settings of group rings and *-algebras. In Chapter 3, we review these techniques and discuss how they relate to the study of non-local games. We also provide a weak sum of squares property for the ring of integers. We show that if a Hermitian element is positive under all unitary representations then it must be expressible as a sum of Hermitian squares.

A copy of the thesis can be found here:  Thesis_Version_3 (1)-1

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