Theory-independent topics towards quantum mechanics

: ψ-ontology and nonlocality distillation

Abstract

It is widely accepted that the most emblematic quantity of quantum mechanics, the wavefunction, is the successful tool for making accurate predictions about experimental results. However, the physical meaning of the wavefunction has been a subject of debates since the birth of the theory. A popular dilemma has to do with the question of whether a pure quantum state is an observer-dependent-quantity/a state-of-knowledge, such as probabilities and density operators, or otherwise if it is something (characterised as) real. Reasons why this debate is important include resolutions of the wavefunction collapse mechanism as well as of potential conflicts between quantum mechanics and special relativity. Physicists have had opinions on this dilemma based on intuitions, interpretations or philosophy. But in the recent years, the community of quantum foundations has been equipped with theorems -known as ψ-ontology theorems- that answer this dilemma rigorously. As always, the “advantage” of a theorem is that it comes with a proof and the “disadvantage” that it comes with assumptions. The common ground among all ψ-ontology theorems is that they work within the framework of underlying models.
In a previous work R. Colbeck and R. Renner constructed a theorem concluded that a quantum state is not a state-of-knowledge about an underlying physical state. A core assumption behind their proof was that the inputs used for a chained Bell test were uncorrelated with all the variables of the setup that are not caused by them. This free-choice definition of them implies no-conspiracy and parameter-independence (in Bell's theorem terms), and actually, the two latter were the strictly necessary for their proof. At the first part of this thesis we relax these assumptions and we test the robustness of the theorem’s conclusion. First, regarding no-conspiracy relaxation, we allow the inputs to be correlated with the underlying variables and through a randomness amplification technique we show that there exist conditions under which the original theorem’s conclusion still holds. The situation is different regarding parameter-dependence embedding, since their proof turns out to be unreceptive to it. Under this evidence, we discuss a particular toy underlying model as a counterexample explaining at least why our intended parameter-independence relaxation would be impossible to reach the maximum parameter-dependence strength.
Non-signalling correlations play a central role in the next parts of the thesis. The realisation by the community that correlations stronger than any quantum ones, can still be non-signalling, created two -among plenty- broad research programs. Since the non-signalling principle cannot single out quantum correlations, is there any other physical axiom managing so? And, when it comes to performing information-processing tasks, how quantum theory competes with other possible non-signalling theories?
Nonlocality distillation, that covers the second part of the thesis, is an absolute “ally” of the earliest aforementioned axiomatization realm. This is because there are correlations that do not violate “at first sight” information-theoretic axioms (such as non-trivial communication complexity, information-causality etc), but they do so after they undergo a distillation process. In a nonlocality distillation situation, we are given a number of nonlocal resources (boxes) characterised by weak nonlocality (according to some measure), and we apply to them operations regarded as free in order to create a final box having stronger nonlocality. However, even in the simplest distillation scenario, that uses two copies of a resource taken from the simplest Bell scenario, there are 82^4 candidate distillation protocols to computationally search over. This fact has been responsible for the extremely slow progress in the whole field. In this work, we initially tackle this chronic obstacle by developing techniques based on Linear Programming and on convex geometry, that significantly speed the search for protocols up. Then, we applied those techniques to construct Serially-Adaptive-Algorithms, algorithms that find out effective ways (in terms of highly amplifying the CHSH measure of nonlocality) to iterate protocols and to combine boxes in a two-by-two manner. After that, we study the almost unexplored so far territory of three-copy CHSH distillation protocols. Overall, four such protocols have been invented in this thesis, that outperform previously known protocols in certain aspects. Three of them are genuine in the sense that they are not reducible to iterations of lower scale (i.e. two-copy) protocols. Our genuine protocols unlock the distillability within sets of quantum correlations as well, that were not known to be distillable before. Every nonlocality distillation scheme developed in this thesis broadens the known set of post-quantum correlations that trivialize communication complexity. Through this prism, each one contributes on ruling out more post-quantum correlations.
The third part of the thesis concerns the generalised probabilistic theory involving the full non-signalling state space (termed boxworld). It was known that while the states of boxworld can be richer, “more entangled” than quantum ones, its measurements are much poorer, following a regime of a “trade-off the measurements for the states”. In this part we shift the attention towards boxworld measurements, and we are asking, even if they are not as elaborate as quantum ones can be, what is the most interesting set of measurements boxworld gives rise to? To this purpose, we characterise the deterministic effects in a certain boxworld scenario, by a method introduced here as an alternative to vertex-enumeration. We distinguish the couplers, the effects that cannot be expressed as trivial operations connecting inputs and outputs between boxes (the latter are wirings). Finally, having those couplers characterised, we spark-off the research field of finding information-processing tasks that benefit from them and indeed, we show that boxworld state-discrimination and nonlocality-distillation are among such tasks.

Date of Award	18 May 2023
Original language	English
Awarding Institution	University of Bristol
Supervisor	Roger Colbeck (Supervisor) & Dara McCutcheon (Supervisor)