Compressibility of positive semidefinite factorizations and quantum models

We investigate compressibility of the dimension of positive semidefinite matrices while approximately preserving their pairwise inner products. This can either be regarded as compression of positive semidefinite factorizations of nonnegative matrices or (if the matrices are subject to additional normalization constraints) as compression of quantum models. We derive both lower and upper bounds on compressibility. Applications are broad and range from the statistical analysis of experimental data to bounding the one-way quantum communication complexity of Boolean functions.Comment: 13 page

Testing classical and quantum theory with single photons

To date, quantum theory is the most successful physical theory that has been discovered. However, there is still the possibility that an inconsistency between experimental observations and the predictions of quantum theory may one day be found, thus prompting the replacement of quantum theory with a superior, post-quantum theory. To narrow down the scope of possibilities for the true theory that describes nature, one can perform experiments that falsify other physical theories, by demonstrating an incompatibility between experimental observations and the predictions of these theories. This thesis details two such experiments. First we provide relevant background information, beginning with a review of experimental quantum optics. Next, we review noncontextual ontological theories and discuss requirements for experimental tests of such theories. Finally, we discuss the framework of generalised probabilistic theories before introducing the two experiments. The first experiment is a test of noncontextual ontological (or hidden-variable) models of nature. An ontological model of a physical theory is one in which systems have preexisting properties, and a noncontextual ontological model is one in which systems that are indistinguishable experimentally are represented identically in the model. Physical theories that cannot be represented by a noncontextual ontological model are said to be nonclassical; quantum theory is an example of such a physical theory that is nonclassical in this sense. Prior to this thesis, experimental tests of the assumption of noncontextuality had assumed that the experiments were free of both systematic and statistical errors, which is not justifiable for any experiment. We introduce new analytical techniques that allow us to avoid making these assumptions, and perform an experiment with single photons that, with high confidence, rules out the possibility of describing nature with a noncontextual ontological model. The second experiment is a demonstration of self-consistent state and measurement tomography in the framework of generalised probabilistic theories (GPTs). The GPT framework is a very general, operationally-motivated framework for describing a physical theory in terms of the observable events predicted by the theory. We develop a technique for inferring the GPT description of a set of states and measurements directly from experimental data. By analysing our data in this general framework, we are able to test various candidate physical theories of nature. We perform an experiment with single photons, and quantify the size of possible variations between quantum theory and the true physical theory that describes nature