A Topos for Algebraic Quantum Theory Heunen, Chris; Landsman, Nicolaas P.; Spitters, Bas
Communications in mathematical physics,
10/2009, Letnik:
291, Številka:
1
Journal Article
Recenzirano
Odprti dostop
The aim of this paper is to relate algebraic quantum mechanics to topos theory, so as to construct new foundations for quantum logic and quantum spaces. Motivated by Bohr’s idea that the empirical ...content of quantum physics is accessible only through classical physics, we show how a
noncommutative
C*-algebra of observables
A
induces a topos
in which the amalgamation of all of its commutative subalgebras comprises a single
commutative
C*-algebra
. According to the constructive Gelfand duality theorem of Banaschewski and Mulvey, the latter has an internal spectrum
in
, which in our approach plays the role of the quantum phase space of the system. Thus we associate a locale (which is the topos-theoretical notion of a space and which intrinsically carries the intuitionistic logical structure of a Heyting algebra) to a C*-algebra (which is the noncommutative notion of a space). In this setting, states on
A
become probability measures (more precisely, valuations) on
, and self-adjoint elements of
A
define continuous functions (more precisely, locale maps) from
to Scott’s interval domain. Noting that open subsets of
correspond to propositions about the system, the pairing map that assigns a (generalized) truth value to a state and a proposition assumes an extremely simple categorical form. Formulated in this way, the quantum theory defined by
A
is essentially turned into a classical theory, internal to the topos
.
These results were inspired by the topos-theoretic approach to quantum physics proposed by Butterfield and Isham, as recently generalized by Döring and Isham.
We relate notions of complementarity in three layers of quantum mechanics: (i) von Neumann algebras, (ii) Hilbert spaces, and (iii) orthomodular lattices. Taking a more general categorical ...perspective of which the above are instances, we consider dagger monoidal kernel categories for (ii), so that (i) become (sub)endohomsets and (iii) become subobject lattices. By developing a ‘point-free’ definition of copyability we link (i) commutative von Neumann subalgebras, (ii) classical structures, and (iii) Boolean subalgebras.
Tensor topology Enrique Moliner, Pau; Heunen, Chris; Tull, Sean
Journal of pure and applied algebra,
October 2020, 2020-10-00, Letnik:
224, Številka:
10
Journal Article
Recenzirano
Odprti dostop
A subunit in a monoidal category is a subobject of the monoidal unit for which a canonical morphism is invertible. They correspond to open subsets of a base topological space in categories such as ...those of sheaves or Hilbert modules. We show that under mild conditions subunits endow any monoidal category with a kind of topological intuition: there are well-behaved notions of restriction, localisation, and support, even though the subunits in general only form a semilattice. We develop universal constructions completing any monoidal category to one whose subunits universally form a lattice, preframe, or frame.
TENSOR-RESTRICTION CATEGORIES Heunen, Chris; Lemay, Jean-Simon Pacaud
Theory and applications of categories,
01/2021, Letnik:
37, Številka:
21
Journal Article
Recenzirano
Restriction categories were established to handle maps that are partially defined with respect to composition. Tensor topology realises that monoidal categories have an intrinsic notion of space, and ...deals with objects and maps that are partially defined with respect to this spatial structure. We introduce a construction that turns a firm monoidal category into a restriction category and axiomatise the monoidal restriction categories that arise this way, called tensor-restriction categories.
We study internal structures in regular categories using monoidal methods. Groupoids in a regular Goursat category can equivalently be described as special dagger Frobenius monoids in its monoidal ...category of relations. Similarly, connectors can equivalently be described as Frobenius structures with a ternary multiplication. We study such ternary Frobenius structures and the relationship to binary ones, generalising that between connectors and groupoids.
Rig groupoids provide a semantic model of Π, a universal classical reversible programming language over finite types. We prove that extending rig groupoids with just two maps and three equations ...about them results in a model of quantum computing that is computationally universal and equationally sound and complete for a variety of gate sets. The first map corresponds to an 8th root of the identity morphism on the unit 1. The second map corresponds to a square root of the symmetry on 1+1. As square roots are generally not unique and can sometimes even be trivial, the maps are constrained to satisfy a nondegeneracy axiom, which we relate to the Euler decomposition of the Hadamard gate. The semantic construction is turned into an extension of Π, called √Π, that is a computationally universal quantum programming language equipped with an equational theory that is sound and complete with respect to the Clifford gate set, the standard gate set of Clifford+T restricted to ≤2 qubits, and the computationally universal Gaussian Clifford+T gate set.
Any associative bilinear multiplication on the set of n-by-n matrices over some field of characteristic not two, that makes the same vectors orthogonal and has the same trace as ordinary matrix ...multiplication, must be ordinary matrix multiplication or its opposite.
Any functor from the category of C*-algebras to the category of locales that assigns to each commutative C*-algebra its Gelfand spectrum must be trivial on algebras of nxn-matrices for n at least 3. ...The same obstruction applies to the Zariski, Stone, and Pierce spectra. The possibility of spectra in categories other than that of locales is briefly discussed.
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