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 $C^{\ast}$ -measure spaces.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 $C^{\ast}$-measure spaces.

Dec 18, 20140 citations

Quantum measures and spaces

Computers & Mathematics with ApplicationsJul 1, 19996 citations

Quantum Measures and Spaces

Gudrun Kalmbach

Springer eBooksJan 1, 199815 citations

Title: Adaptive Topology of Quantum Measurement Spaces (ATQM)

Jincheng Zhang

Quantum measurement protocols are inherently complex, requiring careful consideration of the measurement space to maximize information gained. Traditional approaches often assume a fixed measurement space, limiting the potential for optimization. This paper introduces Adaptive Topology of Quantum Measurement Spaces (ATQM), a novel protocol that dynamically reshapes the quantum state based on machine learning, leading to significant improvements in information extraction from quantum measurements. This adaptive approach offers a more intelligent and flexible framework for measurement design, addressing limitations of existing protocols and opening new avenues for quantum computation.Quantum measurement protocols are inherently complex, requiring careful consideration of the measurement space to maximize information gained. Traditional approaches often assume a fixed measurement space, limiting the potential for optimization. This paper introduces Adaptive Topology of Quantum Measurement Spaces (ATQM), a novel protocol that dynamically reshapes the quantum state based on machine learning, leading to significant improvements in information extraction from quantum measurements. This adaptive approach offers a more intelligent and flexible framework for measurement design, addressing limitations of existing protocols and opening new avenues for quantum computation.

Sep 6, 20260 citations

Finite Quantum Measure Spaces

Stanley Gudder

Quantum measure spaces possess a certain “quantum weirdness” and lack some of the simplicity and intuitive nature of their classical counterparts. Much of this unusual behavior is due to a phenomenon called quantum interference, which is a recurrent theme in the present article. Because of this interference, quantum measures need not be additive but satisfy a more general condition called grade-2 additivity. Examples of quantum measure spaces such as “quantum coins” and particle-an tip article pairs are considered. Even more general spaces called super-quantum measure spaces are discussed. You don't need quantum mechanics or measure theory to understand this article.Quantum measure spaces possess a certain “quantum weirdness” and lack some of the simplicity and intuitive nature of their classical counterparts. Much of this unusual behavior is due to a phenomenon called quantum interference, which is a recurrent theme in the present article. Because of this interference, quantum measures need not be additive but satisfy a more general condition called grade-2 additivity. Examples of quantum measure spaces such as “quantum coins” and particle-an tip article pairs are considered. Even more general spaces called super-quantum measure spaces are discussed. You don't need quantum mechanics or measure theory to understand this article.

American Mathematical MonthlyJan 1, 201016 citations

Quantum Measure Theory

Jan Hamhalter

Springer eBooksJan 1, 2003190 citations

C^{\ast}

-measure spaces.