862 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
Pictures of complete positivity in arbitrary dimension
Two fundamental contributions to categorical quantum mechanics are presented.
First, we generalize the CP-construction, that turns any dagger compact
category into one with completely positive maps, to arbitrary dimension.
Second, we axiomatize when a given category is the result of this construction.Comment: Final versio
Quantum Picturalism
The quantum mechanical formalism doesn't support our intuition, nor does it
elucidate the key concepts that govern the behaviour of the entities that are
subject to the laws of quantum physics. The arrays of complex numbers are kin
to the arrays of 0s and 1s of the early days of computer programming practice.
In this review we present steps towards a diagrammatic `high-level' alternative
for the Hilbert space formalism, one which appeals to our intuition. It allows
for intuitive reasoning about interacting quantum systems, and trivialises many
otherwise involved and tedious computations. It clearly exposes limitations
such as the no-cloning theorem, and phenomena such as quantum teleportation. As
a logic, it supports `automation'. It allows for a wider variety of underlying
theories, and can be easily modified, having the potential to provide the
required step-stone towards a deeper conceptual understanding of quantum
theory, as well as its unification with other physical theories. Specific
applications discussed here are purely diagrammatic proofs of several quantum
computational schemes, as well as an analysis of the structural origin of
quantum non-locality. The underlying mathematical foundation of this high-level
diagrammatic formalism relies on so-called monoidal categories, a product of a
fairly recent development in mathematics. These monoidal categories do not only
provide a natural foundation for physical theories, but also for proof theory,
logic, programming languages, biology, cooking, ... The challenge is to
discover the necessary additional pieces of structure that allow us to predict
genuine quantum phenomena.Comment: Commissioned paper for Contemporary Physics, 31 pages, 84 pictures,
some colo
Hidden measurements, hidden variables and the volume representation of transition probabilities
We construct, for any finite dimension , a new hidden measurement model
for quantum mechanics based on representing quantum transition probabilities by
the volume of regions in projective Hilbert space. For our model is
equivalent to the Aerts sphere model and serves as a generalization of it for
dimensions . We also show how to construct a hidden variables scheme
based on hidden measurements and we discuss how joint distributions arise in
our hidden variables scheme and their relationship with the results of Fine.Comment: 23 pages, 1 figur
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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