Tsirelson's Bound and Beyond: Verifiability and Complexity in Quantum Systems

Abstract

This thesis employs operator-algebraic and group-theoretical techniques to study verifiability and complexity in bipartite quantum systems. A bipartite Bell scenario consists of two non-interacting parties, each can make several quantum measurements. If the two parties share an entangled quantum state, their measurement outcomes can be correlated in surprising ways. In general, we do not directly observe the entangled state and measurement operators (which are referred to as a quantum model), only the resulting statistics (which are referred to as a "correlation") --- there are typically many different models achieving a given correlation. Hence it is remarkable that some correlation has 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 abstract 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 propose an operator-algebraic formulation of robust self-testing. For many nonlocal games of interest, including synchronous games and XOR games, their optimal strategies correspond to tracial states on the associated game algebras. We show that for such nonlocal games, our operator-algebraic definition of robust self-testing is equivalent to the standard one. This, in turn, yields an implication from the uniqueness of tracial states on C*-algebras to robust self-testing for nonlocal games. To address how to compute the robustness function of a self-test explicitly, we provide an enhanced version of a well-known stability result due to Gowers and Hatami and show how it completes a common argument used in self-testing.

Self-testing provides a powerful tool for verifying quantum computations. Given that reliable cloud quantum computers are becoming closer to reality, the concept of verifiability of delegated quantum computations is of central interest. Many models have been proposed, each with specific strengths and weaknesses. In the second part of this thesis, we put forth a new model where the client trusts only its classical processing, makes no computational assumptions, and interacts with a quantum server in a single round. In addition, during a set-up phase, the client specifies the size n of the computation and receives an untrusted, off-the-shelf (OTS) device that is used to report the outcome of a single measurement. We show how to delegate polynomial-time quantum computations in the OTS model. This also yields an interactive proof system for all of QMA, which, furthermore, we show can be accomplished in statistical zero-knowledge. This provides the first relativistic (one-round), two-prover zero-knowledge proof systems for QMA.

Mathematically, bipartite quantum measurement systems can be modeled by the tensor product of free *-algebras. The third part of this thesis studies the complexity of determining positivity of noncommutative polynomials in these algebras. An element of a *-algebra is said to be positive if it is non-negative in all *-representations. In many situations, we'd like to be able to decide whether an element is positive, and if it is, find a certificate of positivity. For noncommutative algebras, it is well known that an element of the free *-algebra is positive if and only if it is a sum of squares. This provides an effective way to determine if a given noncommutative *-polynomial is positive, by searching through sums of squares decompositions. We show that no such procedure exists for the tensor product of two free *-algebras: determining whether a *-polynomial of such an algebra is positive is coRE-hard. We also show that it is coRE-hard to determine whether a noncommutative *-polynomial is trace-positive. Our results hold if free *-algebras are replaced by other algebras that model quantum measurements, such as group algebras of free groups or free products of cyclic groups.