Resumen de Geometría de espacios de banach y aplicaciones en información cuántica
The present thesis develops at the intersection of functional analysis and quantum information science. In this vein, this work is mainly concerned with the study of Position Based Quantum Cryptography from the perspective of local Banach space theory. This programme is conducted first by building a connection between both fields and then, by developing tools in the context of the latter to tackle open questions in the cryptographic setting. Additionally, some of the techniques that we build along the way are of interest to other problems within quantum information. In particular, here we make notorious progress in the understanding of Programmable Quantum Processors.
The questions we approach can be encompassed in the study of resources in quantum information processing. Our main results consist of bounds on the dimension of quantum systems required to perform certain tasks. We obtain results in two different areas. First, in the setting of Programmable Quantum Processors, we investigate the optimal dimension of the memory that is necessary to achieve universal programmability. Our bounds significantly improve the understanding of these objects, exponentially reducing the gap between previously known lower and upper bounds. Second, in the context of Position Based Cryptography, our aim is to explore the amount of entanglement that allows a team of cheaters to break any cryptographic security in this scenario. We obtain new lower bounds on this quantity, which have strong implications for cheaters that operate with enough regularity - in a specific sense defined in the thesis -. We also consider dropping out the regularity assumption and relate the validity of some bounds with open problems in local Banach space theory - the estimation of type constants of tensor norms of finite dimensional Hilbert spaces -.
Indeed, the theory of type and cotype of Banach spaces plays a crucial role in the development of our results. After building connections between the problems we are interested in and the theory of Banach spaces, the study of type constants of certain spaces provides the right framework to obtain the bounds that conform the main results of this thesis. In the case of Universal Programmable Quantum Processors, we achieve a characterization of these objects as nearly isometric embeddings between particular finite dimensional Banach spaces. The type-2 constant of the spaces involved restricts the dimension of these spaces compatible with the existence of such embeddings. In the case of Position Based Cryptography, we study cheating strategies as vector-valued functions on the Boolean cube. A Sobolev-type inequality of Pisier provides the link with type constants. Beyond the results achieved in this thesis, the connections that we build open new avenues that are worth of further exploration. They allow the introduction of new tools in the study of quantum information and provide motivation to explore some constructions and questions in pure mathematics.
