Quantum Foundations

Randomness is a valuable resource in both classical and quantum networks and we wish to generate desired probability distributions as cheaply as possible. If we are allowed to slightly change the distribution under some tolerance level, we can sometimes greatly reduce the cardinality of the randomness or the dimension of the entanglement. By studying statistical inequalities, we show how to upper bound of the amount of randomness required for any given classical network and tolerance level. We also present a problem we encounter when compressing the randomness in a quantum network.

Canonical quantization is not well suited to quantize gravity, while affine quantization is. For those unfamiliar with affine quantization the talk will include a primer. This procedure is then applied to deal with three nonrenormalizable, field theoretical, problems of increasing difficulty, the last one being general relativity itself.

The need for a time-shift invariant formulation of quantum theory arises from fundamental symmetry principles as well as heuristic cosmological considerations. Such a description then leaves open the question of how to reconcile global invariance with the perception of change, locally. By introducing relative time observables, we are able to make rigorous the Page-Wootters conditional probability formalism to show how local Heisenberg evolution is compatible with global invariance.

We derive Born’s rule and the density-operator formalism for quantum systems with Hilbert spaces of finite dimension. Our extension of Gleason’s theorem only relies upon the consistent assignment of probabilities to the outcomes of projective measurements and their classical mixtures. This assumption is significantly weaker than those required for existing Gleason-type theorems valid in dimension two.

The theory of relativity associates a proper time with each moving object via its spacetime trajectory. In quantum theory on the other hand, such trajectories are forbidden. I will discuss an operation approach to exploring this conflict, considering the average time measured by a quantum clock in the weak-field, low-velocity limit. Considering the role of the clock’s state of motion, one finds that all ``good'' quantum clocks experience the time dilation prescribed by general relativity for the most classical states of motion. For nonclassical states of motion, on the other hand, one finds that quantum interference effects give rise to a discrepancy between the proper time and the time measured by the clock. I will also describe how ignorance of the clock's state of motion leads to a larger uncertainty in the time as measured by the clock, a consequence of entanglement between the clock time and its center-of-mass degrees of freedom.

A common criticism directed against many-world theories is that, being deterministic, they cannot make sense of probability. I argue that, on the contrary, deterministic theories with branching provide us the only known coherent definition of objective probability. I illustrate this argument with a toy many-worlds theory known as Kent's universe, and discuss its limitations when applied to the usual Many-Worlds interpretation of quantum mechanics.

I'll also argue that subjective probabilities are unproblematic in the many-worlds setting by showing how the usual decision-theoretical axioms apply there, and finish by showing that together with a proper definition of measurement they suffice to derive the Born rule.

Epistemic interpretations of quantum theory maintain that quantum states only represent incomplete information about the physical states of the world. A major motivation for this view is the promise to provide a reasonable account of state update under measurement by asserting that it is simply a natural feature of updating incomplete statistical information. Here we demonstrate that all known epistemic ontological models of quantum theory in dimension d ≥ 3, including those designed to evade the conclusion of the PBR theorem, cannot represent state update correctly. Conversely, interpretations for which the wavefunction is real evade such restrictions despite remaining subject to long-standing criticism regarding physical discontinuity, indeterminism and the ambiguity of the Heisenberg cut. This revives the possibility of a no-go theorem with no additional assumptions, and demonstrates that what is usually thought of as a strength of epistemic interpretations may in fact be a weakness. We also discuss hidden Markov models and their relationship to ontological models, demarcating the ways in which one might move ‘outside’ the ontological models formalism.

The theory of statistical comparison was formulated (chiefly by David Blackwell in the 1950s) in order to extend the theory of majorization to objects beyond probability distributions, like multivariate statistical models and stochastic transitions, and has played an important role in mathematical statistics ever since. The central concept in statistical comparison is the so-called "information ordering," according to which information need not always be a totally ordered quantity, but often takes on a multi-faceted form whose content may vary depending on its use. In this talk, after reviewing the basic ideas of statistical comparison with an emphasis on their operational character, I will discuss various generalizations to quantum theory (and beyond). I will then argue that quantum statistical comparison provides a natural framework, somehow complementary to semi-definite programming, to study quantum resource theories, with explicit examples given by the resource theories of quantum nonlocality, quantum communication, and quantum thermodynamics.

Group representations are ubiquitous in quantum information theory. Many important states or channels are invariant under particular symmetries: for example depolarizing channels, Werner states, isotropic states, GHZ states. Accordingly, computations involving those objects can be simplified by invoking the symmetries of the problem. For that purpose, we need to know which irreducible representations appear in the problem, and how. Decomposing a representation is a hard problem; however, we can cheat and use numerical techniques to approximate the change of basis matrix -- and even recover exact results.

Recently, a new formulation of quantum mechanics was suggested which is based on the evolution of classical particles, provided with a sign, rather than standard wave functions. This allows several advantages over other approaches: from a theoretical perspective, it offers a more intuitive framework while, from a numerical point of view, it allows the simulation of complex systems with relatively small computational resources. In this talk, I will first go through the tenets of this new approach. In particular, I will focus on the derivation of such theory and the peculiar view it provides in the passage from the quantum to the classical regime. Then, I will discuss the various applications which have been performed so far, especially for systems of Fermions. Finally, a list of possible future works will be presented and discussed.

In this talk I will set out two new contributions to the study of operational tasks in a relativistic quantum setting. First, I will present a generalisation of the task known as ‘summoning,’ in which an unknown quantum state is supplied to an agent and must be returned at a specified point when a corresponding call is made. I will show that when this task is generalised to allow for more than one call to be made, an apparent paradox arises: the extra freedom makes it strictly harder to complete the task. Second, I will describe a quantum generalisation of the classic cryptographic task known as ‘zero-knowledge-proving.’ I will show that there exists no perfectly secure quantum relativistic protocol for this task, and I will then set out a protocol which is conjectured to be close to optimal in security for this task. These results have practical applications for distributed quantum computing and cryptography and also interesting implications for our understanding of relativistic quantum information and its localisation in spacetime.