Quantum Foundations

The aim of this talk is to give an introduction to modular categories, touching both basics and recent developments. I will begin with a quick reminder concerning tensor categories, in particular braided and symmetric ones, and notions like duality, fusion and spherical categories. We'll meet algebras in tensor categories, categories of modules, module categories and their connection. I will then focus on modular categories and some basic structure theory. We will consider two ways of obtaining modular categories: modularization and the Drinfeld center. (The important third one, quantum groups at root of unity, is too complicated to be discussed in any depth.) The Drinfeld center will be used to define Witt equivalence of modular categories and the Witt group. Several equivalent characterization of Witt equivalence, using module categories, will be discussed. The Witt group will be crucial for any (future) classification of modular categories, as well as for application to physics in condensed matter physics and conformal field theory.

I will give an introduction to the Kitaev quantum double models for Hopf C*-algebras. To this end I will introduce a graphical tensor-network notation to represent the algebraic objects and axioms. Using this notation I will then present the vertex- and plaquette symmetries of the model and discuss their interaction and the excitation structure they give rise to.

This talk will be a short introduction to the semisimple Hopf algebras over an algebraically closed field of characteristic 0 and their representation theories. It is intended to outline the main basic results about structure and known methods for the construction of semisimple Hopf algebras: extensions, twisting, Tannakian reconstruction. Basic notions concerning tensor categories will be introduced: braided structures, center construction, fiber functors. Special emphasis is given to the notion of fusion category. At the same time, the relations between these notions and those of the Hopf algebras are studied.

The purpose of this talk is twofold: one, to acquaint the wider community working mostly on Bell-Kochen-Specker contextuality with recent work on Spekkens’ contextuality that quantitatively demonstrates the sense in which Bell-Kochen-Specker contextuality is subsumed within Spekkens’ approach, and two, to argue that one can test for contextuality without appealing to a notion of sharpness which can needlessly restrict the scope of operational theories that could be considered as candidate explanations of experimental data. Testing contextuality in Spekkens’ approach therefore extends the range of experimental scenarios in which contextuality can be witnessed, and refines what it means to witness contextuality in the presence of inevitable noise in KS-type experiments. We will see this for both KS-uncolourability based logical contradiction type proofs of the KS theorem a la Kochen-Specker and statistical proofs on KS-colourable scenarios a la KCBS or Yu-Oh. While Bell-KS contextuality can be mathematically understood as an instance of the classical marginal problem, the same is not true of Spekkens' contextuality. The latter reduces to the classical marginal problem only under very specific conditions, being more general otherwise. All in all, we will argue that all you really need is Leibniz, i.e. identity of indiscernables, to make sense of contextuality in the most general context.

In order to perform foundational experiments testing the correctness of quantum mechanics, one requires data analysis tools that do not assume quantum theory. We introduce a quantum-free tomography technique that fits experimental data to a set of states and measurement effects in a generalised probabilistic theory (GPT). (This is in contrast to quantum tomography, which fits data to sets of density operators and POVM elements.) We perform an experiment on the polarization degree of freedom of single photons, and find GPT descriptions of the states and measurements in our experiment. We gather data for a large number of preparation and measurement procedures in order to map out the spaces of allowed GPT states and measurement effects, and we bound their possible deviation from quantum theory. Our GPT tomography method allows us to bound the extent to which nature might be more or less contextual than quantum theory, as measured by the maximum achievable violation of a particular noncontextuality inequality. We find that the maximal violation is confined to lie between 1.2±0.1% less than and 1.3±0.1% greater than the quantum prediction. Coauthors: Matthew Pusey, Robert Spekkens, Kevin Resch

This talk will be about constraints on any model which reproduces the qubit stabilizer sub-theory. We show that the minimum number of classical bits required to specify the state of an n-qubit system must scale as ~ n(n-3)/2 in any model that does not contradict the predictions of the quantum stabilizer sub-theory. The Gottesman-Knill algorithm, which is a strong simulation algorithm is in fact, very close to this bound as it scales at ~n(2n+1). This is a result of state-independent contextuality which puts a lower bound on the minimum number of states a model requires in order to reproduce the statistics of the qubit stabilizer sub-theory.