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

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

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.

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

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.

Paper Structure

This paper contains 13 sections, 13 theorems, 126 equations.

Table of Contents

  1. Introduction
  2. Operator algebras and the category of non-commutative probability spaces
  3. States and the GNS construction
  4. States and the GNS construction
  5. The modular operator of a state
  6. Completely-positive unital maps
  7. The category of non-commutative probability spaces
  8. Fields of covariances
  9. Classification of continuous fields of covariances in finite dimensions
  10. Tracial states
  11. Non-tracial states
  12. A zoo of continuous fields of covariances in finite dimensions
  13. Conclusions and future work

Key Result

Lemma 1

Let $\{\rho_{n}\}_{n\in\mathbb{N}}$ be a sequence of faithful states on the finite-dimensional $C^*$-algebra $\mathscr{A}$ such that $\|\rho_{n} -\rho\|_{\mathscr{A}^{*}}\to 0$, and let $\mathbf{p}$ be the support projection of $\rho$. Let $\tau$ be a fixed tracial state on $\mathscr{A}$, and let $\

Theorems & Definitions (37)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Definition 1: The category $\mathsf{NCP}$ of non-commutative spaces
  • Definition 2: Split monomorphisms
  • Definition 3: Field of covariances
  • Remark 1: On the choice of morphism action
  • Remark 2
  • Definition 4: Commuting sequence for $\rho$
  • ...and 27 more