Topos quantum mechanics, developed by Döring (
2008
); Döring and Harding Houston J. Math.
42
(2), 559–568 (
2016
); Döring and Isham (
2008
); Flori
2013
)); Flori (
2018
); Isham and Butterfield J. ...Theoret. Phys.
37
, 2669–2733 (
1998
); Isham and Butterfield J. Theoret. Phys.
38
, 827–859 (
1999
); Isham et al. J. Theoret. Phys.
39
, 1413–1436 (
2000
); Isham and Butterfield J. Theoret. Phys.
41
, 613–639 (
2002
), creates a topos of presheaves over the poset
V
(
N
)
of Abelian von Neumann subalgebras of the von Neumann algebra
N
of bounded operators associated to a physical system, and established several results, including: (a) a connection between the Kochen-Specker theorem and the non-existence of a global section of the spectral presheaf; (b) a version of the spectral theorem for self-adjoint operators; (c) a connection between states of
N
and measures on the spectral presheaf; and (d) a model of dynamics in terms of
V
(
N
)
. We consider a modification to this approach using not the whole of the poset
V
(
N
)
, but only its elements
V
(
N
)
∗
of height at most two. This produces a different topos with different internal logic. However, the core results (a)–(d) established using the full poset
V
(
N
)
are also established for the topos over the smaller poset, and some aspects simplify considerably. Additionally, this smaller poset has appealing aspects reminiscent of projective geometry.
Active lattices determine AW-algebras Heunen, Chris; Reyes, Manuel L.
Journal of mathematical analysis and applications,
08/2014, Letnik:
416, Številka:
1
Journal Article
Recenzirano
Odprti dostop
We prove that operator algebras that have enough projections are completely determined by those projections, their symmetries, and the action of the latter on the former. This includes all von ...Neumann algebras and all AW*-algebras. We introduce active lattices, which are formed from these three ingredients. More generally, we prove that the category of AW*-algebras is equivalent to a full subcategory of active lattices. Crucial ingredients are an equivalence between the category of piecewise AW*-algebras and that of piecewise complete Boolean algebras, and a refinement of the piecewise algebra structure of an AW*-algebra that enables recovering its total structure.
The Category of Matroids Heunen, Chris; Patta, Vaia
Applied categorical structures,
04/2018, Letnik:
26, Številka:
2
Journal Article
Recenzirano
Odprti dostop
The structure of the category of matroids and strong maps is investigated: it has coproducts and equalizers, but not products or coequalizers; there are functors from the categories of graphs and ...vector spaces, the latter being faithful and having a nearly full Kan extension; there is a functor to the category of geometric lattices, that is nearly full; there are various adjunctions and free constructions on subcategories, inducing a simplification monad; there are two orthogonal factorization systems; some, but not many, combinatorial constructions from matroid theory are functorial. Finally, a characterization of matroids in terms of optimality of the greedy algorithm can be rephrased in terms of limits.
Mixed quantum states in higher categories Heunen, Chris; Vicary, Jamie; Wester, Linde
Electronic proceedings in theoretical computer science,
12/2014, Letnik:
172, Številka:
Proc. QPL 2014
Journal Article
Odprti dostop
There are two ways to describe the interaction between classical and quantum information categorically: one based on completely positive maps between Frobenius algebras, the other using symmetric ...monoidal 2-categories. This paper makes a first step towards combining the two. The integrated approach allows a unified description of quantum teleportation and classical encryption in a single 2-category, as well as a universal security proof applicable simultaneously to both scenarios.
We introduce a construction that turns a category of pure state spaces and operators into a category of observable algebras and superoperators. For example, it turns the category of ...finite-dimensional Hilbert spaces into the category of finite-dimensional C*-algebras and completely positive maps. In particular, the new category contains both quantum and classical channels, providing elegant abstract notions of preparation and measurement. We also consider nonstandard models that can be used to investigate which notions from algebraic quantum information theory are operationally justifiable.
Higher-order probabilistic programming languages allow programmers to write sophisticated models in machine learning and statistics in a succinct and structured way, but step outside the standard ...measure-theoretic formalization of probability theory. Programs may use both higher-order functions and continuous distributions, or even define a probability distribution on functions. But standard probability theory does not handle higher-order functions well: the category of measurable spaces is not cartesian closed. Here we introduce quasi-Borel spaces. We show that these spaces: form a new formalization of probability theory replacing measurable spaces; form a cartesian closed category and so support higher-order functions; form a well-pointed category and so support good proof principles for equational reasoning; and support continuous probability distributions. We demonstrate the use of quasi-Borel spaces for higher-order functions and probability by: showing that a well-known construction of probability theory involving random functions gains a cleaner expression; and generalizing de Finetti's theorem, that is a crucial theorem in probability theory, to quasi-Borel spaces.
LIMITS IN DAGGER CATEGORIES Heunen, Chris; Karvonen, Martti
Theory and applications of categories,
01/2019, Letnik:
34, Številka:
18
Journal Article
Recenzirano
We develop a notion of limit for dagger categories, that we show is suitable in the following ways: it subsumes special cases known from the literature; dagger limits are unique up to unitary ...isomorphism; a wide class of dagger limits can be built from a small selection of them; dagger limits of a fixed shape can be phrased as dagger adjoints to a diagonal functor; dagger limits can be built from ordinary limits in the presence of polar decomposition; dagger limits commute with dagger colimits in many cases.
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