Generalized Gleason and Kraus Theorems for hybrid classical-quantum probabilities

• URL: http://arxiv.org/abs/2408.10882v1
• Date: Tue, 20 Aug 2024 14:13:44 GMT
• Title: Generalized Gleason and Kraus Theorems for hybrid classical-quantum probabilities
• Authors: S. Camalet,
• Abstract summary: We propose axioms for hybrid classical-quantum probability measures that readily generalize the usual ones for classical and quantum probability measures. We formulate a requirement for the transformations of hybrid probability measures analogous to the complete positive assumption for quantum operations. Explicit expressions for these transformations are derived when the classical and quantum subsystems are non-interacting, the classical subsystem is discrete, or the Hilbert space of the quantum subsystem is finite-dimensional.
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• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We propose four axioms for hybrid classical-quantum probability measures that readily generalize the usual ones for classical and quantum probability measures. A generalized Gleason theorem that gives the mathematical form of the corresponding hybrid states is shown. This form simplifies when the classical subsystem probabilities are described by a probability density function with respect to a natural reference measure. We formulate a requirement for the transformations of hybrid probability measures analogous to the complete positive assumption for quantum operations. For hybrid systems with reference measure, we prove a generalized Kraus theorem that fully determines the considered transformations provided they are continuous with respect to an appropriate metric. Explicit expressions for these transformations are derived when the classical and quantum subsystems are non-interacting, the classical subsystem is discrete, or the Hilbert space of the quantum subsystem is finite-dimensional. We also discuss the quantification of the correlations between the classical and quantum subsystems and a generalization of the quantum operations usually considered in the study of quantum entanglement.

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