7,446 research outputs found
Phase groups and the origin of non-locality for qubits
We describe a general framework in which we can precisely compare the
structures of quantum-like theories which may initially be formulated in quite
different mathematical terms. We then use this framework to compare two
theories: quantum mechanics restricted to qubit stabiliser states and
operations, and Spekkens's toy theory. We discover that viewed within our
framework these theories are very similar, but differ in one key aspect - a
four element group we term the phase group which emerges naturally within our
framework. In the case of the stabiliser theory this group is Z4 while for
Spekkens's toy theory the group is Z2 x Z2. We further show that the structure
of this group is intimately involved in a key physical difference between the
theories: whether or not they can be modelled by a local hidden variable
theory. This is done by establishing a connection between the phase group, and
an abstract notion of GHZ state correlations. We go on to formulate precisely
how the stabiliser theory and toy theory are `similar' by defining a notion of
`mutually unbiased qubit theory', noting that all such theories have four
element phase groups. Since Z4 and Z2 x Z2 are the only such groups we conclude
that the GHZ correlations in this type of theory can only take two forms,
exactly those appearing in the stabiliser theory and in Spekkens's toy theory.
The results point at a classification of local/non-local behaviours by finite
Abelian groups, extending beyond qubits to finitary theories whose observables
are all mutually unbiased.Comment: 24 pages, many picture
Groupoid Semantics for Thermal Computing
A groupoid semantics is presented for systems with both logical and thermal
degrees of freedom. We apply this to a syntactic model for encryption, and
obtain an algebraic characterization of the heat produced by the encryption
function, as predicted by Landauer's principle. Our model has a linear
representation theory that reveals an underlying quantum semantics, giving for
the first time a functorial classical model for quantum teleportation and other
quantum phenomena.Comment: We describe a groupoid model for thermodynamic computation, and a
quantization procedure that turns encrypted communication into quantum
teleportation. Everything is done using higher category theor
Relating toy models of quantum computation: comprehension, complementarity and dagger mix autonomous categories
Toy models have been used to separate important features of quantum
computation from the rich background of the standard Hilbert space model.
Category theory, on the other hand, is a general tool to separate components of
mathematical structures, and analyze one layer at a time. It seems natural to
combine the two approaches, and several authors have already pursued this idea.
We explore *categorical comprehension construction* as a tool for adding
features to toy models. We use it to comprehend quantum propositions and
probabilities within the basic model of finite-dimensional Hilbert spaces. We
also analyze complementary quantum observables over the category of sets and
relations. This leads into the realm of *test spaces*, a well-studied model. We
present one of many possible extensions of this model, enabled by the
comprehension construction. Conspicuously, all models obtained in this way
carry the same categorical structure, *extending* the familiar dagger compact
framework with the complementation operations. We call the obtained structure
*dagger mix autonomous*, because it extends mix autonomous categories, popular
in computer science, in a similar way like dagger compact structure extends
compact categories. Dagger mix autonomous categories seem to arise quite
naturally in quantum computation, as soon as complementarity is viewed as a
part of the global structure.Comment: 21 pages, 6 figures; Proceedings of Quantum Physics and Logic, Oxford
8-9 April 200
A complete graphical calculus for Spekkens' toy bit theory
While quantum theory cannot be described by a local hidden variable model, it
is nevertheless possible to construct such models that exhibit features
commonly associated with quantum mechanics. These models are also used to
explore the question of {\psi}-ontic versus {\psi}-epistemic theories for
quantum mechanics. Spekkens' toy theory is one such model. It arises from
classical probabilistic mechanics via a limit on the knowledge an observer may
have about the state of a system. The toy theory for the simplest possible
underlying system closely resembles stabilizer quantum mechanics, a fragment of
quantum theory which is efficiently classically simulable but also non-local.
Further analysis of the similarities and differences between those two theories
can thus yield new insights into what distinguishes quantum theory from
classical theories, and {\psi}-ontic from {\psi}-epistemic theories.
In this paper, we develop a graphical language for Spekkens' toy theory.
Graphical languages offer intuitive and rigorous formalisms for the analysis of
quantum mechanics and similar theories. To compare quantum mechanics and a toy
model, it is useful to have similar formalisms for both. We show that our
language fully describes Spekkens' toy theory and in particular, that it is
complete: meaning any equality that can be derived using other formalisms can
also be derived entirely graphically. Our language is inspired by a similar
graphical language for quantum mechanics called the ZX-calculus. Thus Spekkens'
toy bit theory and stabilizer quantum mechanics can be analysed and compared
using analogous graphical formalisms.Comment: Major revisions for v2. 22+7 page
Depicting qudit quantum mechanics and mutually unbiased qudit theories
We generalize the ZX calculus to quantum systems of dimension higher than
two. The resulting calculus is sound and universal for quantum mechanics. We
define the notion of a mutually unbiased qudit theory and study two particular
instances of these theories in detail: qudit stabilizer quantum mechanics and
Spekkens-Schreiber toy theory for dits. The calculus allows us to analyze the
structure of qudit stabilizer quantum mechanics and provides a geometrical
picture of qudit stabilizer theory using D-toruses, which generalizes the Bloch
sphere picture for qubit stabilizer quantum mechanics. We also use our
framework to describe generalizations of Spekkens toy theory to higher
dimensional systems. This gives a novel proof that qudit stabilizer quantum
mechanics and Spekkens-Schreiber toy theory for dits are operationally
equivalent in three dimensions. The qudit pictorial calculus is a useful tool
to study quantum foundations, understand the relationship between qubit and
qudit quantum mechanics, and provide a novel, high level description of quantum
information protocols.Comment: In Proceedings QPL 2014, arXiv:1412.810
Survey of mathematical foundations of QFT and perturbative string theory
Recent years have seen noteworthy progress in the mathematical formulation of
quantum field theory and perturbative string theory. We give a brief survey of
these developments. It serves as an introduction to the more detailed
collection "Mathematical Foundations of Quantum Field Theory and Perturbative
String Theory".Comment: This is the introduction to the upcoming volume "Mathematical
Foundations of Quantum Field Theory and Perturbative String Theory", edited
by the authors and published by the American Mathematical Societ
A universe of processes and some of its guises
Our starting point is a particular `canvas' aimed to `draw' theories of
physics, which has symmetric monoidal categories as its mathematical backbone.
In this paper we consider the conceptual foundations for this canvas, and how
these can then be converted into mathematical structure. With very little
structural effort (i.e. in very abstract terms) and in a very short time span
the categorical quantum mechanics (CQM) research program has reproduced a
surprisingly large fragment of quantum theory. It also provides new insights
both in quantum foundations and in quantum information, and has even resulted
in automated reasoning software called `quantomatic' which exploits the
deductive power of CQM. In this paper we complement the available material by
not requiring prior knowledge of category theory, and by pointing at
connections to previous and current developments in the foundations of physics.
This research program is also in close synergy with developments elsewhere, for
example in representation theory, quantum algebra, knot theory, topological
quantum field theory and several other areas.Comment: Invited chapter in: "Deep Beauty: Understanding the Quantum World
through Mathematical Innovation", H. Halvorson, ed., Cambridge University
Press, forthcoming. (as usual, many pictures
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