Quantum Foundations

The distillation of quantum resources such as entanglement and coherence forms one of the most fundamental protocols in quantum information and is of outstanding operational significance, but it is often characterized in the idealized asymptotic limit where an unbounded number of independent and identically distributed copies of a quantum system are available. Physical considerations necessarily limit the number of copies available as well as restrict our ability to perform coherent state manipulations over a large numbers of systems, which makes it crucial to be able to characterize how well we can distill resources in realistic, non-asymptotic settings.

We introduce a framework for the non-asymptotic characterization of two related operational tasks, the distillation as well as environment-assisted distillation of quantum coherence. We establish a complete description of the achievable rates of distillation in the one-shot setting under several different classes of free operations, which we show to correspond to the optimization of smoothed entropic quantities as semidefinite programs. We introduce a class of coherence measures which quantify the best achievable fidelity of distillation, and use them to obtain an explicit analytical characterization of coherence distillation for all pure states. Further, we provide insight into the distillation of entanglement, revealing new operational similarities and differences between the resource theories of coherence and entanglement.

This talk is based on joint work with Kun Fang, Xin Wang, and Gerardo Adesso (arxiv:1711.10512) as well as with Ludovico Lami and Alexander Streltsov (in preparation).

The methodology employed in reconstructing quantum theory involves defining a general mathematical framework that frames a landscape of possible theories and then positing principles that uniquely pick out quantum theory. In contrast, many traditional interpretations of quantum theory consider only quantum theory, not a larger space of possible theories. I will defend the modal methodology used in reconstruction by tracing the historical roots of Einstein’s distinction between principle and constructive theories. Einstein’s principle theories (exemplified by thermodynamics and special relativity) are often cited as inspiration for reconstructing quantum theory. The concept of a “physics of principles” emerged at the end of the nineteenth century in the context of the application of Lagrangian mechanics to electromagnetism. This case is an intriguing historical precedent for reconstructing quantum theory. I will also offer some reflections on how the application of a similar modal methodology in axiomatic QFT plays out. 

Hidden-variables theories account for quantum mechanics in terms of a particular 'equilibrium' distribution of underlying parameters corresponding to the Born rule. In the most well-studied example, the pilot-wave theory of de Broglie and Bohm, it is well established that the Born rule may be understood to arise from a process of dynamical relaxation. This 'quantum relaxation' may have taken place in the very early universe and could have left imprints on the cosmic microwave background (CMB). Such imprints amount to signatures of the decay of early violations of the Born rule. In this colloquium we summarise recent progress in making detailed predictions and in comparing them with the reported large-scale anomalies in the CMB data.

Hidden-variables theories account for quantum mechanics in terms of a particular 'equilibrium' distribution of underlying parameters corresponding to the Born rule. A natural question to ask is whether the theory is stable under small perturbations away from equilibrium. We compare and contrast two examples: de Broglie's 1927 pilot-wave theory and Bohm's 1952 reformulation thereof. It is well established that in de Broglie's dynamics initial deviations from equilibrium will relax. We show that this is not the case for Bohm's dynamics: initial deviations from equilibrium do not relax and in fact grow with time. On this basis we argue that Bohm's dynamics is untenable as a physical theory (while de Broglie's dynamics remains a viable candidate). We advocate stability as a general selection criterion for hidden-variables theories.

In this talk I show how to systematically classify all possible alternatives to the measurement postulates of quantum theory. All alternative measurement postulates are in correspondence with a representation of the unitary group. I will discuss composite systems in these alternative theories and show that they violate two operational properties: purification and local tomography. This shows that one can derive the measurement postulates of quantum theory from either of these properties. I will discuss the relevance of this result to the field of general probabilistic theories. In a second part of the talk I will discuss work in progress and directions for future research. I will show how to generalise the framework used to theories which have different pure states and dynamics than quantum theory. I will discuss two types of theories which can be studied in this framework: Grassmannian theories (same dynamical group and different pure states to quantum theory) and non-linear modifications to the Schrodinger equation (same pure states and different dynamical group).

Science is like puzzle-solving. Making sense of quantum theory is a particularly thorny kind of brain-twister, with more than its fair share of mysteries. If you are stuck on a puzzle, it may be because you have made a false assumption about the nature of some entity that is absolutely central to the whole business. If so, you have made a category mistake: you are not just wrong about what this entity is, but about what sort of thing it is.

In his Public Lecture at Perimeter Institute, Robert Spekkens will explain why he believes that many quantum mysteries are a result of a category mistake concerning the nature of quantum states. Along the way, he will address some idiosyncratic questions, such as: What did Plato have to say about Heisenberg’s uncertainty principle? What do poorly implemented clinical drug trials have to do with "spooky action at a distance"? And, most importantly, what did the successful deciphering of Egyptian hieroglyphs teach us about the interpretation of quantum theory? 

Spekkens is a faculty member at Perimeter Institute whose research examines the foundations of quantum theory. He co-edited the book Quantum Theory: Informational Foundations and Foils, and he is a Project Leader of the international research collaboration "Quantum Causal Structures.” In 2012, he won first prize in the Foundational Questions Institute (FQXi) essay contest "Questioning the Foundations: Which of Our Assumptions Are Wrong?" He lives in Waterloo with his wife and three-year-old son.