Experimentally Testable Noncontextuality Inequalities Via Fourier-Motzkin Elimination

Abstract

Generalized noncontextuality as defined by Spekkens is an attempt to make precise the distinction between classical and quantum theories. It is a restriction on models used to reproduce the predictions of quantum mechanics. There has been considerable progress recently in deriving experimentally robust noncontextuality inequalities. The violation of these inequalities is a sufficient condition for an experiment to not admit a generalized noncontextual model. In this thesis, we present an algorithm to automate the derivation of noncontextuality inequalities. At the heart of this algorithm is a technique called Fourier Motzkin elimination (abbrev. FM elimination). After a brief overview of the generalized notion of contextuality and FM elimination, we proceed to demon- strate how this algorithm works by using it to derive noncontextuality inequalities for a number of examples. Belinfante’s construction, presented first for the sake of pedagogy, is based on a simple proof by Belinfante to demonstrate the invalidity of von Neumann’s notion of noncontextuality. We provide a quantum realization on a single qubit that can violate this inequality. We then go on to discuss paradigmatic proofs of noncontextuality such as the Peres-Mermin square, Mermin’s Star and the 18 vector construction and use them to derive robust noncontextuality inequalities using the algorithm. We also show how one can generalize the noncontextuality inequalities obtained for the Peres-Mermin square to systems with continuous variables.