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Open Quantum Systems Decoherence Quantum Simulation

Fields of covariances on non-commutative probability spaces in finite dimensions

arXiv
Authors: Florio M. Ciaglia, Fabio Di Cosmo, Laura González-Bravo

Year

2025

Paper ID

17961

Status

Preprint

Abstract Read

~2 min

Abstract Words

104

Citations

N/A

Abstract

We introduce the notion of a field of covariances, a contravariant functor from non-commutative probability spaces to Hilbert spaces, as the natural categorical analogue of statistical covariance. In the case of finite-dimensional non-commutative probability spaces, we obtain a complete classification of such fields. Our results unify classical and quantum information geometry: in the tracial case, we recover (a contravariant version of) Cencov's uniqueness of the Fisher-Rao metric, while in the faithful case, we recover (a contravariant version of) the Morozova-Cencov-Petz classification of quantum monotone metrics. Crucially, our classification extends naturally to non-faithful states that are not pure, thus generalizing Petz and Sudar's radial extension.

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  • We introduce the notion of a field of covariances, a contravariant functor from non-commutative probability spaces to Hilbert spaces, as the natural categorical analogue of...

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References & Citation Signals

[1] DOI https://doi.org/arXiv:2510.24617 [2] arXiv https://arxiv.org/abs/2510.24617 [3] Publisher https://arxiv.org/abs/2510.24617

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