27 research outputs found
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
Complete Positivity for Mixed Unitary Categories
In this article we generalize the \CP^\infty-construction of dagger
monoidal categories to mixed unitary categories. Mixed unitary categories
provide a setting, which generalizes (compact) dagger monoidal categories and
in which one may study quantum processes of arbitrary (infinite) dimensions.
We show that the existing results for the \CP^\infty-construction hold in
this more general setting. In particular, we generalize the notion of
environment structures to mixed unitary categories and show that the
\CP^\infty-construction on mixed unitary categories is characterized by this
generalized environment structure.Comment: Lots of figure
Groupoids, Frobenius algebras and Poisson sigma models
In this paper we discuss some connections between groupoids and Frobenius
algebras specialized in the case of Poisson sigma models with boundary. We
prove a correspondence between groupoids in the category Set and relative
Frobenius algebras in the category Rel, as well as an adjunction between a
special type of semigroupoids and relative H*-algebras. The connection between
groupoids and Frobenius algebras is made explicit by introducing what we called
weak monoids and relational symplectic groupoids, in the context of Poisson
sigma models with boundary and in particular, describing such structures in the
ex- tended symplectic category and the category of Hilbert spaces. This is part
of a joint work with Alberto Cattaneo and Chris Heunen.Comment: 12 pages, 1 figure. To appear in "Mathematical Aspects of Quantum
Field Theories". Mathematical Physical Studies, Springer. Proceedings of the
Winter School in Mathematical Physics, Les Houges, 201
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
Bases as Coalgebras
The free algebra adjunction, between the category of algebras of a monad and
the underlying category, induces a comonad on the category of algebras. The
coalgebras of this comonad are the topic of study in this paper (following
earlier work). It is illustrated how such coalgebras-on-algebras can be
understood as bases, decomposing each element x into primitives elements from
which x can be reconstructed via the operations of the algebra. This holds in
particular for the free vector space monad, but also for other monads, like
powerset or distribution. For instance, continuous dcpos or stably continuous
frames, where each element is the join of the elements way below it, can be
described as such coalgebras. Further, it is shown how these
coalgebras-on-algebras give rise to a comonoid structure for copy and delete,
and thus to diagonalisation of endomaps like in linear algebra