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arXiv:1412.5756

Classical and quantum conditional measures from a categorical viewpoint

Hitoshi Motoyama, Kohei Tanaka

Published Dec 18, 2014• 1 citation
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Abstract

This paper presents categorical structures on classical measure spaces and quantum measure spaces in order to deal with canonical maps associated with conditional measures as morphisms. We extend the Riesz-Markov-Kakutani representation theorem and the Gelfand duality theorem to an equivalence of categories between them. From this categorical viewpoint, we introduce a quantum version of conditional measures as a dual concept of the classical one.

DOI
10.48550/arxiv.1412.5756

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This paper presents categorical structures on classical measure spaces and quantum measure spaces in order to deal with canonical maps associated with conditional measures as morphisms. We extend the Riesz-Markov-Kakutani representation theorem and the Gelfand duality theorem to an equivalence of categories between them. From this categorical viewpoint, we introduce a quantum version of conditional measures as a dual concept of the classical one.
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.
Markov categories have recently emerged as a powerful high-level framework for probability theory and theoretical statistics. Here we study a quantum version of this concept, called involutive Markov categories. These are equivalent to Parzygnat's quantum Markov categories, but we argue that they offer a simpler and more practical approach. Our main examples of involutive Markov categories have pre-C*-algebras, including infinite-dimensional ones, as objects, together with completely positive unital maps as morphisms in the picture of interest. In this context, we prove a quantum de Finetti theorem for both the minimal and the maximal C*-tensor norms, and we develop a categorical description of such quantum de Finetti theorems which amounts to a universal property of state spaces.
The most general type of measurement in quantum physics is modeled by a positive operator-valued measure (POVM). Mathematically, a POVM is a generalization of a measure, whose values are not real numbers, but positive operators on a Hilbert space. POVMs can equivalently be viewed as maps between effect algebras or as maps between algebras for the Giry monad. We will show that this equivalence is an instance of a duality between two categories. In the special case of continuous POVMs, we obtain two equivalent representations in terms of morphisms between von Neumann algebras.
It is well known that classical and quantum theories carry distinct types of representations, each type of representation corresponding to possible values of generalized charges in the classical or quantum context. This paper demonstrates a sense in the structure of these representation theories is preserved from classical to quantum physics. To show this, I discuss distinct representation-theory preserving morphisms in the classical and quantum contexts. Specifically, I consider categories whose morphisms are Lagrangian relations in the classical context and Hilbert bimodules in the quantum context. These morphisms are significant because they give rise to induced representations of classical and quantum theories, respectively. I consider quantization and the classical limit as determining functors between these categories. I treat quantization via the strict deformation quantization of a Poisson algebra and the classical limit via the extension of a uniformly continuous bundle of C*-algebras. With these tools, I prove that the quantization and classical limit functors are "almost-inverse" to each other, thus establishing a categorical equivalence.
This paper proposes a comprehensive categorical framework for unifying classical and quantum probabilistic logics. We leverage the inherent compositional and structural advantages of category theory to construct a common mathematical language capable of expressing both Kolmogorovian probability and the non-Boolean structures characteristic of quantum mechanics. By treating systems and processes as objects and morphisms within suitable monoidal categories, we demonstrate how core concepts like states, observables, measurements, and conditioning can be abstractly represented and seamlessly transitioned between classical and quantum domains. The framework utilizes probability monads for classical uncertainty and dagger compact closed categories for quantum processes, showing how these distinct structures can be seen as instances of a more general categorical paradigm. This unification not only provides a deeper foundational understanding of uncertainty across different physical theories but also opens new avenues for developing hybrid probabilistic-logical systems in fields such as artificial intelligence and quantum computation.