Quantum Foundations

In this talk I am going to describe Spekkens’ toy model, a non-contextual hidden variable model with an epistemic restriction, a constraint on what an observer can know about reality. The aim of the model, developed for continuous and discrete prime degrees of freedom, is to advocate the epistemic view of quantum theory, where quantum states are states of incomplete knowledge about a deeper underlying reality. In spite of its classical flavour, many aspects that were thought to belong only to quantum mechanics can be reproduced in the model. 

I am going to describe our results regarding the formulation of rules for the update of states after measurement. I will do it for systems of discrete prime dimensions and I will then give the idea on how to proceed in the non-prime dimensional case. 

I will also depict the relationship between Spekkens’ model, stabiliser quantum mechanics and Gross' theory of discrete Wigner functions (they are equivalent theories in odd dimensions) in terms of measurement update rules.

I will conclude by briefly discussing a project we have been recently working on that consists of characterising sub theories of Spekkens’ model that are operationally equivalent to sub theories of QM (in particular in the case of qubits) and use them to represent the non-contextual classically simulable part of state-injection schemes of computation with contextuality as a resource.

What is the essence of quantum theory? In the present talk I want to approach this question from a particular operationalist perspective. I take advantage of a recent convergence between operational approaches to quantum theory and axiomatic approaches to quantum field theory. Removing anything special to particular physical models, including underlying notions of space and (crucially) time, what remains is what I shall call "abstract quantum theory". This embeds into a more general framework that also includes classical theory, with classical and quantum theory representing two extremes in a spectrum of possible theories. I shall present this within a "hierarchy of abstraction", where adding structure leads to more specialized frameworks. In particular, adding topological spacetime and locality leads to the recently proposed positive formalism and recovers quantum field theory. Further rigidifying time and adding causality recovers the standard formulation of quantum theory. There are other promising specializations such as Hardy's proposal to work in op-space.

What can we say about the spectra of a collection of microscopic variables when only their coarse-grained sums are experimentally accessible? In this paper, using the tools and methodology from the study of quantum nonlocality, we develop a mathematical theory of the macroscopic fluctuations generated by ensembles of independent microscopic discrete systems. We provide algorithms to decide which multivariate gaussian distributions can be approximated by sums of finitely-valued random vectors. We study non-trivial cases where the microscopic variables have an unbounded range, as well as asymptotic scenarios with infinitely many macroscopic variables. From a foundational point of view, our results imply that bipartite gaussian states of light cannot be understood as beams of independent d-dimensional particle pairs. It is also shown that the classical description of certain macroscopic optical experiments, as opposed to the quantum one, requires variables with infinite cardinality spectra.

Computational complexity theory is a branch of computer science dedicated to classifying computational problems in terms of their difficulty. While computability theory tells us what we can compute in principle, complexity theory informs us regarding what is feasible. In this chapter I argue that the science of quantum computing illuminates the foundations of complexity theory by emphasising that its fundamental concepts are not model-independent. However this does not, as some have suggested, force us to radically revise the foundations of the theory. For model-independence never has been essential to those foundations. The fundamental aim of complexity theory is to describe what is achievable in practice under various models of computation for our various practical purposes. Reflecting on quantum computing illuminates complexity theory by reminding us of this, too often under-emphasised, fact.

Entropy is an important information measure. A complete understanding of entropy flow will have applications in quantum thermodynamics and beyond; for example it may help to identify the sources of fidelity loss in quantum communications and methods to prevent or control them. Being nonlinear in density matrix, its evaluation for quantum systems requires simultaneous evolution of more-than-one density matrix. Recently in [1] a formalism for such an evolution has been proposed and [2] shows that the flow of entropy between two systems corresponds to the full counting statistics of physical quantities that are exchanged between them. Interestingly, in quantum systems with heat dissipations this will not be equivalent to the second law of thermodynamics. In this talk I will describe a consistent formalism to evaluate entropy and show how to measure it in some quantum systems; for example in quantum point contacts in nanoelectronics and in the quantum heat engines introduced to describe photosynthesis and photovoltaic cells. The entropy flow is made of two parts: 1) an incoherent part, which can be re-evaluated semiclassically from the second law, and 2) a coherent part, which has no semiclassical analogue and appear as a result of extending Kubo-Martin-Schwinger (KMS) correlations.

[1] M.H.A. and Y. Nazarov, Phys. Rev. B 91, 104303 (2015)

[2] M.H.A. and Y. Nazarov, Phys. Rev. B 91, 174307 (2015)