Autoparallelity of Quantum Statistical Manifolds in Light of Quantum Estimation Theory

TL;DR

This paper explores the geometric structure of quantum statistical manifolds, focusing on the properties of e-autoparallel submanifolds and their relation to quantum estimation theory, highlighting differences from classical information geometry.

Contribution

It introduces quantum analogues of classical exponential families, characterizes e-autoparallel submanifolds via estimation theory, and discusses mathematical properties of quantum geometric connections.

Findings

e-autoparallel submanifolds relate to efficient quantum estimators

The e-connection has non-zero torsion unlike classical cases

Results extend to more general affine connections in geometry

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.