Semidefinite programming relaxations for quantum correlations

Semidefinite programs are convex optimisation problems involving a linear objective function and a domain of positive semidefinite matrices. Over the last two decades, they have become an indispensable tool in quantum information science. Many otherwise intractable fundamental and applied problems can be successfully approached by means of relaxation to a semidefinite program. Here, we review such methodology in the context of quantum correlations. We discuss how the core idea of semidefinite relaxations can be adapted for a variety of research topics in quantum correlations, including nonlocality, quantum communication, quantum networks, entanglement, and quantum cryptography.Comment: To be submitted to Reviews of Modern Physic

Neural function approximation on graphs: shape modelling, graph discrimination & compression

Graphs serve as a versatile mathematical abstraction of real-world phenomena in numerous scientific disciplines. This thesis is part of the Geometric Deep Learning subject area, a family of learning paradigms, that capitalise on the increasing volume of non-Euclidean data so as to solve real-world tasks in a data-driven manner. In particular, we focus on the topic of graph function approximation using neural networks, which lies at the heart of many relevant methods. In the first part of the thesis, we contribute to the understanding and design of Graph Neural Networks (GNNs). Initially, we investigate the problem of learning on signals supported on a fixed graph. We show that treating graph signals as general graph spaces is restrictive and conventional GNNs have limited expressivity. Instead, we expose a more enlightening perspective by drawing parallels between graph signals and signals on Euclidean grids, such as images and audio. Accordingly, we propose a permutation-sensitive GNN based on an operator analogous to shifts in grids and instantiate it on 3D meshes for shape modelling (Spiral Convolutions). Following, we focus on learning on general graph spaces and in particular on functions that are invariant to graph isomorphism. We identify a fundamental trade-off between invariance, expressivity and computational complexity, which we address with a symmetry-breaking mechanism based on substructure encodings (Graph Substructure Networks). Substructures are shown to be a powerful tool that provably improves expressivity while controlling computational complexity, and a useful inductive bias in network science and chemistry. In the second part of the thesis, we discuss the problem of graph compression, where we analyse the information-theoretic principles and the connections with graph generative models. We show that another inevitable trade-off surfaces, now between computational complexity and compression quality, due to graph isomorphism. We propose a substructure-based dictionary coder - Partition and Code (PnC) - with theoretical guarantees that can be adapted to different graph distributions by estimating its parameters from observations. Additionally, contrary to the majority of neural compressors, PnC is parameter and sample efficient and is therefore of wide practical relevance. Finally, within this framework, substructures are further illustrated as a decisive archetype for learning problems on graph spaces.Open Acces

Guesswork

The security of systems is often predicated on a user or application selecting an object, a password or key, from a large list. If an inquisitor wishing to identify the object in order to gain access to a system can only query each possibility, one at a time, then the number of guesses they must make in order to identify the selected object is likely to be large. If the object is selected uniformly at random using, for example, a cryptographically secure pseudo-random number generator, then the analysis of the distribution of the number of guesses that the inquisitor must make is trivial. If the object has not been selected perfectly uniformly, but with a distribution that is known to the inquisitor, then the quantification of security is relatively involved. This thesis contains contributions to the study of this subject, dubbed Guesswork, motivated both by fundamental investigations into computational security as well as modern applications in secure storage and communication. This thesis begins with two introductory chapters. One describes existing results in Guesswork and summarizes the contributions found in the thesis. The other recapitulates some of the mathematical tools that are employed in the thesis. The other five chapters of contain new contributions to our understanding of Guesswork, much of which has already experienced peer review and been published. The chapters themselves are designed to be self-contained and so readable in isolation

Guesswork

The security of systems is often predicated on a user or application selecting an object, a password or key, from a large list. If an inquisitor wishing to identify the object in order to gain access to a system can only query each possibility, one at a time, then the number of guesses they must make in order to identify the selected object is likely to be large. If the object is selected uniformly at random using, for example, a cryptographically secure pseudo-random number generator, then the analysis of the distribution of the number of guesses that the inquisitor must make is trivial. If the object has not been selected perfectly uniformly, but with a distribution that is known to the inquisitor, then the quantification of security is relatively involved. This thesis contains contributions to the study of this subject, dubbed Guesswork, motivated both by fundamental investigations into computational security as well as modern applications in secure storage and communication. This thesis begins with two introductory chapters. One describes existing results in Guesswork and summarizes the contributions found in the thesis. The other recapitulates some of the mathematical tools that are employed in the thesis. The other five chapters of contain new contributions to our understanding of Guesswork, much of which has already experienced peer review and been published. The chapters themselves are designed to be self-contained and so readable in isolation

State-of-the-art in aerodynamic shape optimisation methods

Aerodynamic optimisation has become an indispensable component for any aerodynamic design over the past 60 years, with applications to aircraft, cars, trains, bridges, wind turbines, internal pipe flows, and cavities, among others, and is thus relevant in many facets of technology. With advancements in computational power, automated design optimisation procedures have become more competent, however, there is an ambiguity and bias throughout the literature with regards to relative performance of optimisation architectures and employed algorithms. This paper provides a well-balanced critical review of the dominant optimisation approaches that have been integrated with aerodynamic theory for the purpose of shape optimisation. A total of 229 papers, published in more than 120 journals and conference proceedings, have been classified into 6 different optimisation algorithm approaches. The material cited includes some of the most well-established authors and publications in the field of aerodynamic optimisation. This paper aims to eliminate bias toward certain algorithms by analysing the limitations, drawbacks, and the benefits of the most utilised optimisation approaches. This review provides comprehensive but straightforward insight for non-specialists and reference detailing the current state for specialist practitioners

Formal Approaches to a Definition of Agents

This thesis is a contribution to the formalisation of the notion of an agent within the class of finite multivariate Markov chains. In accordance with the literature agents are are seen as entities that act, perceive, and are goaldirected. We present a new measure that can be used to identify entities (called i-entities). The intuition behind this is that entities are spatiotemporal patterns for which every part makes every other part more probable. The measure, complete local integration (CLI), is formally investigated within the more general setting of Bayesian networks. It is based on the specific local integration (SLI) which is measured with respect to a partition. CLI is the minimum value of SLI over all partitions. Upper bounds are constructively proven and a possible lower bound is proposed. We also prove a theorem that shows that completely locally integrated spatiotemporal patterns occur as blocks in specific partitions of the global trajectory. Conversely we can identify partitions of global trajectories for which every block is completely locally integrated. These global partitions are the finest partitions that achieve a SLI less or equal to their own SLI. We also establish the transformation behaviour of SLI under permutations of the nodes in the Bayesian network. We then go on to present three conditions on general definitions of entities. These are most prominently not fulfilled by sets of random variables i.e. the perception-action loop, which is often used to model agents, is too restrictive a setting. We instead propose that any general entity definition should in effect specify a subset of the set of all spatiotemporal patterns of a given multivariate Markov chain. Any such definition will then define what we call an entity set. The set of all completely locally integrated spatiotemporal patterns is one example of such a set. Importantly the perception-action loop also naturally induces such an entity set. We then propose formal definitions of actions and perceptions for arbitrary entity sets. We show that these are generalisations of notions defined for the perception-action loop by plugging the entity-set of the perception-action loop into our definitions. We also clearly state the properties that general entity-sets have but the perception-action loop entity set does not. This elucidates in what way we are generalising the perception-action loop. Finally we look at some very simple examples of bivariate Markov chains. We present the disintegration hierarchy, explain it via symmetries, and calculate the i-entities. Then we apply our definitions of perception and action to these i-entities