8 research outputs found
Orthomodular lattices, Foulis Semigroups and Dagger Kernel Categories
This paper is a sequel to arXiv:0902.2355 and continues the study of quantum
logic via dagger kernel categories. It develops the relation between these
categories and both orthomodular lattices and Foulis semigroups. The relation
between the latter two notions has been uncovered in the 1960s. The current
categorical perspective gives a broader context and reconstructs this
relationship between orthomodular lattices and Foulis semigroups as special
instance.Comment: 31 page
Dagger Categories of Tame Relations
Within the context of an involutive monoidal category the notion of a
comparison relation is identified. Instances are equality on sets, inequality
on posets, orthogonality on orthomodular lattices, non-empty intersection on
powersets, and inner product on vector or Hilbert spaces. Associated with a
collection of such (symmetric) comparison relations a dagger category is
defined with "tame" relations as morphisms. Examples include familiar
categories in the foundations of quantum mechanics, such as sets with partial
injections, or with locally bifinite relations, or with formal distributions
between them, or Hilbert spaces with bounded (continuous) linear maps. Of one
particular example of such a dagger category of tame relations, involving sets
and bifinite multirelations between them, the categorical structure is
investigated in some detail. It turns out to involve symmetric monoidal dagger
structure, with biproducts, and dagger kernels. This category may form an
appropriate universe for discrete quantum computations, just like Hilbert
spaces form a universe for continuous computation
A computer scientist's reconstruction of quantum theory
The rather unintuitive nature of quantum theory has led numerous people to
develop sets of (physically motivated) principles that can be used to derive
quantum mechanics from the ground up, in order to better understand where the
structure of quantum systems comes from. From a computer scientist's
perspective we would like to study quantum theory in a way that allows
interesting transformations and compositions of systems and that also includes
infinite-dimensional datatypes. Here we present such a compositional
reconstruction of quantum theory that includes infinite-dimensional systems.
This reconstruction is noteworthy for three reasons: it is only one of a few
that includes no restrictions on the dimension of a system; it allows for both
classical, quantum, and mixed systems; and it makes no a priori reference to
the structure of the real (or complex) numbers. This last point is possible
because we frame our results in the language of category theory, specifically
the categorical framework of effectus theory.Comment: 42 page
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
Dagger and Dilation in the Category of Von Neumann algebras
This doctoral thesis is a mathematical study of quantum computing,
concentrating on two related, but independent topics. First up are dilations,
covered in chapter 2. In chapter 3 "diamond, andthen, dagger" we turn to the
second topic: effectus theory. Both chapters, or rather parts, can be read
separately and feature a comprehensive introduction of their own