Autoparallelity of Quantum Statistical Manifolds in Light of Quantum Estimation Theory

Abstract:In this paper we study the autoparallelity w.r.t. the e-connection for an information-geometric structure called the SLD structure, which consists of a Riemannian metric and mutually dual e- and m-connections, induced on the manifold of strictly positive density operators. Unlike the classical information geometry, the e-connection has non-vanishing torsion, which brings various mathematical difficulties. The notion of e-autoparallel submanifolds is regarded as a quantum version of exponential families in classical statistics, which is known to be characterized as statistical models having efficient estimators (unbiased estimators uniformly achieving the equality in the Cramer-Rao inequality). As quantum extensions of this classical result, we present two different forms of estimation-theoretical characterizations of the e-autoparallel submanifolds. We also give several results on the e-autoparallelity, some of which are valid for the autoparallelity w.r.t. an affine connection in a more general geometrical situation.

Quantum Physics (quant-ph); Differential Geometry (math.DG)

arXiv:2307.03431 [quant-ph]

(or arXiv:2307.03431v3 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2307.03431 Focus to learn more arXiv-issued DOI via DataCite

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