Cirelson Bounds, Rigidity and Robustness of Non-Local Games

Specialeforsvar ved Nicholas Gauguin Houghto-Larsen

Titel: Cirelson Bounds, Rigidity and Robustness of Non-Local Games

Abstract: In this master thesis, we study the notions of rigidity and robustness of non-local quantum games. Rigidity of a game refers to the phenomenon that there exists a strategy for the game achieving an optimal score (the so-called Cirelson bound of the game), such that any other optimal strategy is equivalent, in a certain sense, to that strategy. Robustness refers to the approximate version of that phenomenon, i.e. that
strategies achieving near-optimal scores are close to the pre-specified optimal
strategy. The CHSH-game was historically one of the first to be recognized as
robustly rigid, and it appears in examples throughout. First, a short account
of the formalism of quantum physics is given. Then, we present a mathematical
framework for discussing very general non-local games and quantum strategies
for such games. We give an example of a new kind of game accommodated by this
generality, the `Circle-CHSH-game', which is a continuous version of the
CHSH-game. We propose a new, unified measure of robustness, provide some
theoretical arguments in favor of its usage, and discuss the relation of this
measure to the ad hoc measures prominent in the literature. Then, we present a
new theorem which exerts manageable bounds on the robustness properties of a
very large class of games. Next, we introduce a notion of duality between
certain projection-valued measures and certain unitary representations, which
clarifies in a novel way the especially simple nature of so-called XOR-games,
which are widely studied in the literature. We also indicate how it may be used
to analyze the Circle-CHSH-game. Finally, following \cite{MS}, we discuss
so-called qubit strategies for XOR-games with binary question sets, and extend
the analysis to arbitrary strategies by use of Jordan's lemma. We present a new
proof of Jordan's lemma, based on a theorem by Shapiro. The proof is simple,
very different from the existing one, and suggests that generalizations of
Jordan's lemma are unlikely to exist.

Vejleder:  Matthias Christiandl
Censor:     Wojciek Szymanski, SDU