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Towards a Category-Theoretic Foundation of Classical and Quantum Information Geometry
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DOI：10.1007/978-3-032-03921-7_26.png)
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Abstract
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We introduce the category NCP, whose objects are pairs of W*-algebras and normal states and whose morphisms are state-preserving unital completely positive (CPU) maps, as a common stage for classical and quantum information geometry, and we formulate two results that will appear in forthcoming works. First, we recast the problem of classifying admissible Riemannian geometries on classical and quantum statistical models in terms of functors C: NCP -> Hilb. These functors provide a generalization of classical statistical covariance, and we call them fields of covariances. A prominent example being the so-called GNS functor arising from the Gelfand-Naimark-Segal (GNS) construction. The classification of fields of covariances on NCP entails both Cencov's uniqueness of the Fisher-Rao metric tensor and Petzs classification of monotone quantum metric tensors as particular cases. Then, we show how classical and quantum statistical models can be realized as subcategories of NCP in a way that takes into account symmetries. In this setting, the fields of covariances determine Riemannian metric tensors on the model that reduce to the Fisher-Rao, Fubini-Study, and Bures-Helstrom metric tensor in particular cases.
Keywords:
Information Geometry
Cencov theorem
Operator algebras
Non-commutative probability spaces
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