Fields of covariances on non-commutative probability spaces in finite dimensions
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909873976901632 |
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| author | Ciaglia, Florio M. Di Cosmo, Fabio González-Bravo, Laura |
| author_facet | Ciaglia, Florio M. Di Cosmo, Fabio González-Bravo, Laura |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_24617 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Fields of covariances on non-commutative probability spaces in finite dimensions Ciaglia, Florio M. Di Cosmo, Fabio González-Bravo, Laura Mathematical Physics Quantum Physics 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. |
| title | Fields of covariances on non-commutative probability spaces in finite dimensions |
| topic | Mathematical Physics Quantum Physics |
| url | https://arxiv.org/abs/2510.24617 |