246 research outputs found
An embedding theorem for Hilbert categories
We axiomatically define (pre-)Hilbert categories. The axioms resemble those
for monoidal Abelian categories with the addition of an involutive functor. We
then prove embedding theorems: any locally small pre-Hilbert category whose
monoidal unit is a simple generator embeds (weakly) monoidally into the
category of pre-Hilbert spaces and adjointable maps, preserving adjoint
morphisms and all finite (co)limits. An intermediate result that is important
in its own right is that the scalars in such a category necessarily form an
involutive field. In case of a Hilbert category, the embedding extends to the
category of Hilbert spaces and continuous linear maps. The axioms for
(pre-)Hilbert categories are weaker than the axioms found in other approaches
to axiomatizing 2-Hilbert spaces. Neither enrichment nor a complex base field
is presupposed. A comparison to other approaches will be made in the
introduction.Comment: 24 page
Characterizations of categories of commutative C*-subalgebras
We aim to characterize the category of injective *-homomorphisms between
commutative C*-subalgebras of a given C*-algebra A. We reduce this problem to
finding a weakly terminal commutative subalgebra of A, and solve the latter for
various C*-algebras, including all commutative ones and all type I von Neumann
algebras. This addresses a natural generalization of the Mackey-Piron
programme: which lattices are those of closed subspaces of Hilbert space? We
also discuss the way this categorified generalization differs from the original
question.Comment: 24 page
H*-algebras and nonunital Frobenius algebras: first steps in infinite-dimensional categorical quantum mechanics
A certain class of Frobenius algebras has been used to characterize
orthonormal bases and observables on finite-dimensional Hilbert spaces. The
presence of units in these algebras means that they can only be realized
finite-dimensionally. We seek a suitable generalization, which will allow
arbitrary bases and observables to be described within categorical
axiomatizations of quantum mechanics. We develop a definition of H*-algebra
that can be interpreted in any symmetric monoidal dagger category, reduces to
the classical notion from functional analysis in the category of (possibly
infinite-dimensional) Hilbert spaces, and hence provides a categorical way to
speak about orthonormal bases and quantum observables in arbitrary dimension.
Moreover, these algebras reduce to the usual notion of Frobenius algebra in
compact categories. We then investigate the relations between nonunital
Frobenius algebras and H*-algebras. We give a number of equivalent conditions
to characterize when they coincide in the category of Hilbert spaces. We also
show that they always coincide in categories of generalized relations and
positive matrices.Comment: 29 pages. Final versio
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
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
On discretization of C*-algebras
The C*-algebra of bounded operators on the separable infinite-dimensional
Hilbert space cannot be mapped to a W*-algebra in such a way that each unital
commutative C*-subalgebra C(X) factors normally through .
Consequently, there is no faithful functor discretizing C*-algebras to
AW*-algebras, including von Neumann algebras, in this way.Comment: 5 pages. Please note that arXiv:1607.03376 supersedes this paper. It
significantly strengthens the main results and includes positive results on
discretization of C*-algebra
Noncommutativity as a colimit
Every partial algebra is the colimit of its total subalgebras. We prove this
result for partial Boolean algebras (including orthomodular lattices) and the
new notion of partial C*-algebras (including noncommutative C*-algebras), and
variations such as partial complete Boolean algebras and partial AW*-algebras.
The first two results are related by taking projections. As corollaries we find
extensions of Stone duality and Gelfand duality. Finally, we investigate the
extent to which the Bohrification construction, that works on partial
C*-algebras, is functorial.Comment: 22 pages; updated theorem 15, added propoisition 3
Bohrification
New foundations for quantum logic and quantum spaces are constructed by
merging algebraic quantum theory and topos theory. Interpreting Bohr's
"doctrine of classical concepts" mathematically, given a quantum theory
described by a noncommutative C*-algebra A, we construct a topos T(A), which
contains the "Bohrification" B of A as an internal commutative C*-algebra. Then
B has a spectrum, a locale internal to T(A), the external description S(A) of
which we interpret as the "Bohrified" phase space of the physical system. As in
classical physics, the open subsets of S(A) correspond to (atomic)
propositions, so that the "Bohrified" quantum logic of A is given by the
Heyting algebra structure of S(A). The key difference between this logic and
its classical counterpart is that the former does not satisfy the law of the
excluded middle, and hence is intuitionistic. When A contains sufficiently many
projections (e.g. when A is a von Neumann algebra, or, more generally, a
Rickart C*-algebra), the intuitionistic quantum logic S(A) of A may also be
compared with the traditional quantum logic, i.e. the orthomodular lattice of
projections in A. This time, the main difference is that the former is
distributive (even when A is noncommutative), while the latter is not.
This chapter is a streamlined synthesis of 0709.4364, 0902.3201, 0905.2275.Comment: 44 pages; a chapter of the first author's PhD thesis, to appear in
"Deep Beauty" (ed. H. Halvorson
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