Yuming Zhao, Department of Pure Mathematics, University of Waterloo
“Tsirelson's Bound and Beyond: Verifiability and Complexity in Quantum Systems”
Suppose we have a physical system consisting of two separate labs, each can mark several measurements. If the two labs are entangled, then their measurement statistics can be correlated in surprising ways. In general, we do not directly see the entangled state and measurement operators, only the resulting correlations. There are typically many different models achieving a given correlation, hence it is remarkable that some correlations have a unique quantum model. A correlation with this property is called a self-test. In the first part of this thesis, we give a new definition of self-testing in terms of states on C*-algebras. We show that this operator-algebraic definition of self-testing is equivalent to the standard one and naturally extends to the commuting operator framework for nonlocal correlations. We also give an operator-algebraic formulation of robust self-testing in terms of tracial states on C*-algebras.
Self-testing provides a powerful tool for verifying quantum computations. In the second part of this thesis, we propose a new model of delegated quantum computation where the client trusts only its classical processing and can verify the server's quantum computation, and the server can conceal the inner workings of their quantum devices. This delegation protocol also yields the first two-prover one-round zero-knowledge proof systems of QMA.
Mathematically, bipartite measurements can be modeled by the tensor product of free *-algebras. Many problems for nonlocal correlations are closely related to deciding whether an element of these algebras is positive and finding certificates of positivity. In the third part of this thesis, we show that it is undecidable (coRE-hard) to determine whether a noncommutative polynomial of the tensor product of free *-algebras is positive.
QNC 2101