5 research outputs found
On the Extension of Complex Numbers
This paper proposes an extension of the complex numbers, adding further imaginary units and preserving the idea of the product as a geometric construction. These `supercomplex numbers\u27, denoted S, are studied, and it is found that the algebra contains both new and old phenomena. It is established that equal-dimensional subspaces of S containing R are isomorphic under algebraic operations, whereby a symmetry within the space of imaginary units is illuminated. Certain equations are studied, and also a connection to special relativity is set up and explored. Finally, abstraction leads to the notion of a `generalised supercomplex algebra\u27; both the supercomplex numbers and the quaternions are found to be such algebras
A Mathematical Framework for Causally Structured Dilations and its Relation to Quantum Self-Testing
The motivation for this thesis was to recast quantum self-testing [MY98,MY04]
in operational terms. The result is a category-theoretic framework for
discussing the following general question: How do different implementations of
the same input-output process compare to each other? In the proposed framework,
an input-output process is modelled by a causally structured channel in some
fixed theory, and its implementations are modelled by causally structured
dilations formalising hidden side-computations. These dilations compare through
a pre-order formalising relative strength of side-computations. Chapter 1
reviews a mathematical model for physical theories as semicartesian symmetric
monoidal categories. Many concrete examples are discussed, in particular
quantum and classical information theory. The key feature is that the model
facilitates the notion of dilations. Chapter 2 is devoted to the study of
dilations. It introduces a handful of simple yet potent axioms about dilations,
one of which (resembling the Purification Postulate [CDP10]) entails a duality
theorem encompassing a large number of classic no-go results for quantum
theory. Chapter 3 considers metric structure on physical theories, introducing
in particular a new metric for quantum channels, the purified diamond distance,
which generalises the purified distance [TCR10,Tom12] and relates to the Bures
distance [KSW08a]. Chapter 4 presents a category-theoretic formalism for
causality in terms of '(constructible) causal channels' and 'contractions'. It
simplifies aspects of the formalisms [CDP09,KU17] and relates to traces in
monoidal categories [JSV96]. The formalism allows for the definition of 'causal
dilations' and the establishment of a non-trivial theory of such dilations.
Chapter 5 realises quantum self-testing from the perspective of chapter 4, thus
pointing towards the first known operational foundation for self-testing.Comment: PhD thesis submitted to the University of Copenhagen (ISBN
978-87-7125-039-8). Advised by prof. Matthias Christandl, submitted 1st of
December 2020, defended 11th of February 2021. Keywords: dilations, applied
category theory, quantum foundations, causal structure, quantum self-testing.
242 pages, 1 figure. Comments are welcom
Dilations and information flow axioms in categorical probability
We study the positivity and causality axioms for Markov categories as
properties of dilations and information flow in Markov categories, and in
variations thereof for arbitrary semicartesian monoidal categories. These help
us show that being a positive Markov category is merely an additional property
of a symmetric monoidal category (rather than extra structure). We also
characterize the positivity of representable Markov categories and prove that
causality implies positivity, but not conversely. Finally, we note that
positivity fails for quasi-Borel spaces and interpret this failure as a privacy
property of probabilistic name generation.Comment: 42 page
An operational environment for quantum self-testing
Observed quantum correlations are known to determine in certain cases the
underlying quantum state and measurements. This phenomenon is known as
(quantum) self-testing.
Self-testing constitutes a significant research area with practical and
theoretical ramifications for quantum information theory. But since its
conception two decades ago by Mayers and Yao, the common way to rigorously
formulate self-testing has been in terms of operator-algebraic identities, and
this formulation lacks an operational interpretation. In particular, it is
unclear how to formulate self-testing in other physical theories, in
formulations of quantum theory not referring to operator-algebra, or in
scenarios causally different from the standard one.
In this paper, we explain how to understand quantum self-testing
operationally, in terms of causally structured dilations of the input-output
channel encoding the correlations. These dilations model side-information which
leaks to an environment according to a specific schedule, and we show how
self-testing concerns the relative strength between such scheduled leaks of
information. As such, the title of our paper has double meaning: we recast
conventional quantum self-testing in terms of information-leaks to an
environment -- and this realises quantum self-testing as a special case within
the surroundings of a general operational framework.
Our new approach to quantum self-testing not only supplies an operational
understanding apt for various generalisations, but also resolves some
unexplained aspects of the existing definition, naturally suggests a distance
measure suitable for robust self-testing, and points towards self-testing as a
modular concept in a larger, cryptographic perspective.Comment: Accepted for publication in Quantum. Keywords: quantum information,
quantum computation, quantum self-testing, foundations of physics, causal
structure, dilation