PhD Defense - Nicholas Gauguin Houghton-Larsen
A Mathematical Framework for Causally Structured Dilations andits Relation to Quantum Self-Testing
This is a PhD thesis within the sub-field of mathematical physics that pertains to quantum information theory. Most of its results can be interpreted in the mathematical language of category theory, and may as such be of interest also outside of quantum information theory.
In high-level terms, I present a framework in which one can argue mathematically about aspects of the following fundamental question: How do two given implementations of the same physical process compare to each other? Though of independent interest, the main motivation for this question comes from the area of quantum self-testing ([MY98, MY04]), where one desires to understand all the different ways in which a given set of measurement statistics can be produced by an implementation of local measurements on a multipartite quantum state. The problem which motivated the thesis is that although the traditional envision of quantum self-testing is mathematically precise, the language in which it is cast has no clear operational interpretation.
According to the framework proposed in the thesis, a collection of measurement statistics is regarded as the input-output behaviour of an information channel, and the various implementations of this channel correspond to causally structured computations which may besecretly executed in the environment of the channel during our interaction with it. The maincontribution of the thesis is to introduce a formalism which makes the previous sentence precise, and to provide its relation to the usual definition of quantum self-testing. The relation is essentially that quantum self-testing corresponds to the existence of an implementation from which all others can be derived, and which moreover holds no pre-existing information about the outputs of the channel. This constitutes a first step towards recasting quantum self-testing in purely operational (theory-independent) terms.
Chapter 1 reviews a variation on a category-theoretic model for physical theories. This model includes quantum information theory and classical information theory, but also more mathematical examples such as any category with finite products (e.g. the categories of sets or groups), and any partially ordered commutative monoid, when suitably interpreted. The key feature of the model is that it facilitates the notion of marginals (as known from e.g. classical probability theory), and the dual notion of dilations.
Dilations are the topic of Chapter 2. The results presented there are conceptually independent of quantum self-testing, but rather initiate a systematic study of dilations and constitute an original proof of concept, by demonstrating that several features of information theories can be derived from a handful of principles which reference only the structure of dilations.
Chapter 3 contains some initial thoughts as to how to make an approximate (metric) version of the theory of dilations, and a new metric for quantum channels, the purified diamond distance is introduced. It generalises the purified distance of Refs. [TCR10, Tom12].
Chapter 4 lays out a formalism for arguing about information channels whose outputs are causally contingent on their inputs. This can be seen as a generalised alternative to the framework of quantum combs ([CDP09]), but can also be viewed as generalising the abstract notion of traces in symmetric monoidal categories ([JSV96]). The formalism allows us to make precise the notion of a causal dilation, which captures the above-mentioned causally structured side-computations.
Finally, in Chapter 5, the connection to quantum self-testing is established. This chapter also contains simple proofs of a few general results about self-testing, and a novel recharacterisation of the set of quantum behaviours in terms of non-signalling properties of their Stinespring dilations.
Invitation online defense will be announced later
link to the thesis will be announced later
Matthias Christandl (University of Copenhagen)
Roger Colbeck (University of York, UK)
Tobias Fritz (University of Innsbruck, Austria)
Nathalie Wahl (University of Copenhagen, Denmark)Chair