77 research outputs found
Symmetry, Compact Closure and Dagger Compactness for Categories of Convex Operational Models
In the categorical approach to the foundations of quantum theory, one begins
with a symmetric monoidal category, the objects of which represent physical
systems, and the morphisms of which represent physical processes. Usually, this
category is taken to be at least compact closed, and more often, dagger
compact, enforcing a certain self-duality, whereby preparation processes
(roughly, states) are inter-convertible with processes of registration
(roughly, measurement outcomes). This is in contrast to the more concrete
"operational" approach, in which the states and measurement outcomes associated
with a physical system are represented in terms of what we here call a "convex
operational model": a certain dual pair of ordered linear spaces -- generally,
{\em not} isomorphic to one another. On the other hand, state spaces for which
there is such an isomorphism, which we term {\em weakly self-dual}, play an
important role in reconstructions of various quantum-information theoretic
protocols, including teleportation and ensemble steering. In this paper, we
characterize compact closure of symmetric monoidal categories of convex
operational models in two ways: as a statement about the existence of
teleportation protocols, and as the principle that every process allowed by
that theory can be realized as an instance of a remote evaluation protocol ---
hence, as a form of classical probabilistic conditioning. In a large class of
cases, which includes both the classical and quantum cases, the relevant
compact closed categories are degenerate, in the weak sense that every object
is its own dual. We characterize the dagger-compactness of such a category
(with respect to the natural adjoint) in terms of the existence, for each
system, of a {\em symmetric} bipartite state, the associated conditioning map
of which is an isomorphism
Symmetry and Self-Duality in Categories of Probabilistic Models
This note adds to the recent spate of derivations of the probabilistic
apparatus of finite-dimensional quantum theory from various axiomatic packages.
We offer two different axiomatic packages that lead easily to the Jordan
algebraic structure of finite-dimensional quantum theory. The derivation relies
on the Koecher-Vinberg Theorem, which sets up an equivalence between order-unit
spaces having homogeneous, self-dual cones, and formally real Jordan algebras.Comment: In Proceedings QPL 2011, arXiv:1210.029
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
Local tomography and the Jordan structure of quantum theory
Using a result of H. Hanche-Olsen, we show that (subject to fairly natural
constraints on what constitutes a system, and on what constitutes a composite
system), orthodox finite-dimensional complex quantum mechanics with
superselection rules is the only non-signaling probabilistic theory in which
(i) individual systems are Jordan algebras (equivalently, their cones of
unnormalized states are homogeneous and self-dual), (ii) composites are locally
tomographic (meaning that states are determined by the joint probabilities they
assign to measurement outcomes on the component systems) and (iii) at least one
system has the structure of a qubit. Using this result, we also characterize
finite dimensional quantum theory among probabilistic theories having the
structure of a dagger-monoidal category
Operational theories and Categorical quantum mechanics
A central theme in current work in quantum information and quantum
foundations is to see quantum mechanics as occupying one point in a space of
possible theories, and to use this perspective to understand the special
features and properties which single it out, and the possibilities for
alternative theories. Two formalisms which have been used in this context are
operational theories, and categorical quantum mechanics. The aim of the present
paper is to establish strong connections between these two formalisms. We show
how models of categorical quantum mechanics have representations as operational
theories. We then show how nonlocality can be formulated at this level of
generality, and study a number of examples from this point of view, including
Hilbert spaces, sets and relations, and stochastic maps. The local, quantum,
and no-signalling models are characterized in these terms.Comment: 37 pages, updated bibliograph
Composites and Categories of Euclidean Jordan Algebras
We consider possible non-signaling composites of probabilistic models based
on euclidean Jordan algebras (EJAs), satisfying some reasonable additional
constraints motivated by the desire to construct dagger-compact categories of
such models. We show that no such composite has the exceptional Jordan algebra
as a direct summand, nor does any such composite exist if one factor has an
exceptional summand, unless the other factor is a direct sum of one-dimensional
Jordan algebras (representing essentially a classical system). Moreover, we
show that any composite of simple, non-exceptional EJAs is a direct summand of
their universal tensor product, sharply limiting the possibilities.
These results warrant our focussing on concrete Jordan algebras of hermitian
matrices, i.e., euclidean Jordan algebras with a preferred embedding in a
complex matrix algebra}. We show that these can be organized in a natural way
as a symmetric monoidal category, albeit one that is not compact closed. We
then construct a related category InvQM of embedded euclidean Jordan algebras,
having fewer objects but more morphisms, that is not only compact closed but
dagger-compact. This category unifies finite-dimensional real, complex and
quaternionic mixed-state quantum mechanics, except that the composite of two
complex quantum systems comes with an extra classical bit.
Our notion of composite requires neither tomographic locality, nor
preservation of purity under tensor product. The categories we construct
include examples in which both of these conditions fail. In such cases, the
information capacity (the maximum number of mutually distinguishable states) of
a composite is greater than the product of the capacities of its constituents.Comment: 60 pages, 3 tables. Substantially revised, with some new result
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