Autoparallelity of Quantum Statistical Manifolds in Light of Quantum Estimation Theory
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arXiv
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866912627383336960 |
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| author | Nagaoka, Hiroshi Fujiwara, Akio |
| author_facet | Nagaoka, Hiroshi Fujiwara, Akio |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_03431 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Autoparallelity of Quantum Statistical Manifolds in Light of Quantum Estimation Theory Nagaoka, Hiroshi Fujiwara, Akio Quantum Physics Differential Geometry 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. |
| title | Autoparallelity of Quantum Statistical Manifolds in Light of Quantum Estimation Theory |
| topic | Quantum Physics Differential Geometry |
| url | https://arxiv.org/abs/2307.03431 |