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arXiv:2105.13642 · Symmetry

Category Algebras and States on Categories

Hayato Saigo

Published Jun 29, 2021• 4 citations
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Abstract

The purpose of this paper is to build a new bridge between category theory and a generalized probability theory known as noncommutative probability or quantum probability, which was originated as a mathematical framework for quantum theory, in terms of states as linear functional defined on category algebras. We clarify that category algebras can be considered to be generalized matrix algebras and that the notions of state on category as linear functional defined on category algebra turns out to be a conceptual generalization of probability measures on sets as discrete categories. Moreover, by establishing a generalization of famous GNS (Gelfand–Naimark–Segal) construction, we obtain a representation of category algebras of †-categories on certain generalized Hilbert spaces which we call semi-Hilbert modules over rigs. The concepts and results in the present paper will be useful for the studies of symmetry/asymmetry since categories are generalized groupoids, which themselves are generalized groups.

DOI
10.3390/sym13071172

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We study the mathematical foundations of physics. We reconstruct textbook quantum theory from a single symmetric monoidal functor GNS:Phys⟶∗ModGNS : \mathbf{Phys} \longrightarrow \ast\mathbf{Mod}GNS:Phys⟶∗Mod, based on the Gelfand-Naimark-Segal construction and the notion of representability. We derive the probabilistic interpretation of quantum mechanics, including the Born rule, the Schrödinger and Heisenberg pictures, the relation between symmetries and group representations, and a theory of quantum Markov processes, including wave function collapse. Inclusion of the classical limit and deformation quantization is briefly sketched. Gauge symmetry and extended locality cannot currently be accommodated, due to conceptual difficulties discussed in an appendix.
A quantum set is defined to be simply a set of nonzero finite-dimensional Hilbert spaces. Together with binary relations, essentially the quantum relations of Weaver, quantum sets form a dagger compact category. Functions between quantum sets are certain binary relations that can be characterized in terms of this dagger compact structure, and the resulting category of quantum sets and functions generalizes the category of ordinary sets and functions in the manner of noncommutative mathematics. In particular, this category is dual to a subcategory of von Neumann algebras. The basic properties of quantum sets are presented thoroughly, with the noncommutative dictionary in mind, and with an eye to convenient application. As a motivating example, a notion of quantum graph coloring is derived within this framework, and it is shown to be equivalent to the notion that appears in the quantum information theory literature.
In this paper, the generalized quantum states, i.e. positive and normalized linear functionals on [Formula: see text]-algebras, are studied. First, we study normal states, i.e. states which are represented by density operators, and singular states, i.e. states cannot be represented by density operators. It is given an approach to the resolution of bounded linear functionals into quantum states by applying the Gelfand–Neumark–Segal (GNS) construction, i.e. the fundamental result of Gelfand, Neumark and Segal on the representation theory of [Formula: see text]-algebras, and theory of projections. Second, it is given an application in quantum information theory. We study covariant cloners, i.e. quantum channels in the Heisenberg and the Schrödinger pictures which are covariant by shifting, and it is shown that the optimal cloners cannot have a singular component. Finally, we discuss on the representation of pure states in the sense of the Gelfand–Pettis integral. We also give physical interpretations and examples in different sections of this work.
A quantum set is defined to be simply a set of nonzero finite-dimensional Hilbert spaces. Together with binary relations, essentially the quantum relations of Weaver, quantum sets form a dagger compact category. Functions between quantum sets are certain binary relations that can be characterized in terms of this dagger compact structure, and the resulting category of quantum sets and functions generalizes the category of ordinary sets and functions in the manner of noncommutative mathematics. In particular, this category is dual to a subcategory of von Neumann algebras. The basic properties of quantum sets are presented thoroughly, with the noncommutative dictionary in mind, and with an eye to convenient application. As a motivating example, a notion of quantum graph coloring is derived within this framework, and it is shown to be equivalent to the notion that appears in the quantum information theory literature.
We introduce a concept of conditional measures in quantum measure spaces from a view point of Gelfand duality between classical measure spaces and quantum measure spaces. Classical conditional measure spaces are based on subspaces of a space, hence we define quantum conditional measure spaces based on the dual notion, ideals and quotients of an algebra. Although the classical Bayes rule indicates the ratio of conditional measure (probability) and original one, the quantum Bayes rule describes the difference between them in quantum measure spaces. We also give categorical structures on classical and quantum measure spaces based on Bayes' rule, respectively. These extend the Riesz-Markov-Kakutani representation theorem to an equivalence of categories between the category of classical measure spaces and the category of commutative C∗C^{\ast}C∗-measure spaces.
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.