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

• URL: http://arxiv.org/abs/2510.24617v1
• Date: Tue, 28 Oct 2025 16:45:18 GMT
• Title: Fields of covariances on non-commutative probability spaces in finite dimensions
• Abstract summary: We introduce the notion of a field of covariances, a contravariant functor from non-commutative probability spaces to Hilbert spaces.<n>In the case of finite-dimensional non-commutative probability spaces, we obtain a complete classification of such fields.
• Score: 0.0
• License: http://creativecommons.org/licenses/by-nc-sa/4.0/
• 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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