Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2307.03431 · Information Geometry

Autoparallelity of quantum statistical manifolds in light of quantum estimation theory

Hiroshi Nagaoka, Akio Fujiwara

Published Sep 5, 2026• 0 citations
arXivHTMLPDF

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 Cramér-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 geometric situation.

DOI
10.1007/s41884-026-00217-3

Related papers

Quantum Statistical Manifolds

Jan Naudts

Quantum information geometry studies families of quantum states by means of differential geometry. A new approach is followed with the intention to facilitate the introduction of a more general theory in subsequent work. To this purpose, the emphasis is shifted from a manifold of strictly positive density matrices to a manifold of faithful quantum states on the C*-algebra of bounded linear operators. In addition, ideas from the parameter-free approach to information geometry are adopted. The underlying Hilbert space is assumed to be finite-dimensional. In this way, technicalities are avoided so that strong results are obtained, which one can hope to prove later on in a more general context. Two different atlases are introduced, one in which it is straightforward to show that the quantum states form a Banach manifold, the other which is compatible with the inner product of Bogoliubov and which yields affine coordinates for the exponential connection.

arXiv:1805.10857•Entropy•Jun 17, 2018•17 citations

Submanifolds of Curved Exponential Family in Quantum Statistics

Huafei Sun

Quantum version of the curved exponential family is studied where its dual structure with different metrics.As an important case the Kubo-Mori metric is introduced.The α-connections are set up so that the exponential family is ±1-affine with respect to the Kubo-Mori metric.

Transactions of Beijing Institute of Technology•Jan 1, 2009•0 citations

Duality of quantum geometries

Jan Naudts

Quantum connections are defined by parallel transport operators acting on a Hilbert space. They transport tangent operators along paths in parameter space. The metric tensor of a Riemannian manifold is replaced by an inner product of pairs of operator fields, similar to the inner product of the Kubo-Mori formalism of Linear Response Theory. The metric is used to define the dual of a quantum connection.The gradient of the parallel transport operators is the quantum vector potential. It defines the covariant derivative of operator fields. The covariant derivatives are used to quantify the holonomy of the quantum connection. It is shown that a quantum connection is holonomic if and only if its dual is holonomic. If the parallel transport operators are unitary then an alpha-family of quantum connections can be defined in a way similar to Amari's alpha family of connections in Information Geometry. The minus alpha connection is the dual of the alpha connection. In particular, the alpha equal zero connection is self-dual. An operator field can be combined with a path in parameter space to produce a path in operator space. A definition is given for such a path in operator space to be autoparallel. The path in parameter space is then a geodesic for the induced connection.

arXiv:2401.17908•Jan 31, 2024•0 citations

Information geometry and the hydrodynamical formulation of quantum mechanics

Mathieu Molitor

Let (M,g) be a compact, connected and oriented Riemannian manifold. We denote D the space of smooth probability density functions on M. In this paper, we show that the Frechet manifold D is equipped with a Riemannian metric g^{D} and an affine connection \nabla^{D} which are infinite dimensional analogues of the Fisher metric and exponential connection in the context of information geometry. More precisely, we use Dombrowski's construction together with the couple (g^{D},\nabla^{D}) to get a (non-integrable) almost Hermitian structure on D, and we show that the corresponding fundamental 2-form is a symplectic form from which it is possible to recover the usual Schrodinger equation for a quantum particle living in M. These results echo a recent paper of the author where it is stressed that the Fisher metric and exponential connection are related (via Dombrowski's construction) to Kahler geometry and quantum mechanics in finite dimension.

arXiv:1204.0663•Apr 3, 2012•3 citations

Εφαρμογές της γεωμετρίας της πληροφορίας

Fotios D. Oikonomou, Φώτιος Οικονόμου

Information geometry usually refers to a class of geometries characterized from metrics and corresponding connections related to the notion of statistical information during the process of parameter estimation on classical or quantum systems. We are aware of the fact that in parameter estimation we can't achieve perfectness. There is a lower bound, which is not zero, for the variance - covariance of the estimated parameter compared to the real parameter, the Cramer - Rao bound. In classical parameter estimation, this bound is given by the reciprocal of Fisher information and is unique in the sense that the Fisher is the unique monotone metric under stochastic transformations. In quantum parameter estimation there is a class of monotone metrics and the lowest possible bound, which determines the best quality of estimation, is given by that of the Symmetric Logarithmic Derivative (SLD). The term symmetric determines the derivative type, since there are more than one types due to the non-commutativity of operators, in general, in quantum mechanics. In our first published paper in the framework of this Ph.D. thesis, we are working beyond the class on monotone metrics. We are using the phase space to define a new statistical metric. This is achieved by introducing logarithmic derivatives through the Husimi Q distribution and consequently the new metric. As it turns out the related bound of Cramer - Rao type, can be greater even of this of the Symmetric Logarithmic Derivative. In our second paper, we refer to a particular bi-parameter family of generalized entropies used for the classification of complex systems. By defining a divergence for the discrete distribution, we consequently define a metric and the corresponding Levi - Civita connection, which depend on the two parameters say c and d. So, we are having geometrical objects with statistical content like the Cramer - Rao bound and the scalar curvature, which is characteristic (and different) for every class of complex systems. In our third paper, we are treating the above parameters c and d as coordinates of a two dimensional manifold. By calculating the Fisher metric and the corresponding scalar curvature we observe particular values of c and d for which we are making the conjecture that they related with particular properties of complex systems. Finally, we are highlighting a correlation of the curvature of a particular manifold and quantum entanglement. In particular, we are using the entropy of entanglement to define a two dimensional manifold, which we embed into three dimensional real space. We find that in the region which this surface has maximum curvature we have product quantum states and only there.

•Jan 1, 2019•0 citations

Exponential arcs in manifolds of quantum states

Jan Naudts

The manifold under consideration consists of the faithful normal states on a sigma-finite von Neumann algebra in standard form. Tangent planes and approximate tangent planes are discussed. A relative entropy/divergence function is assumed to be given. It is used to generalize the notion of an exponential arc connecting one state to another. The generator of the exponential arc is shown to be unique up to an additive constant. In the case of Araki’s relative entropy, every self-adjoint element of the von Neumann algebra generates an exponential arc. The generators of the composed exponential arcs are shown to add up. The metric derived from Araki’s relative entropy is shown to reproduce the Kubo–Mori metric. The latter is the metric used in linear response theory. The e- and m-connections describe a dual pair of geometries. Any finite number of linearly independent generators determines a submanifold of states connected to a given reference state by an exponential arc. Such a submanifold is a quantum generalization of a dually flat statistical manifold.

arXiv:2209.12761•Frontiers in Physics•Feb 7, 2023•3 citations
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 Cramér-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 geometric situation.
Quantum information geometry studies families of quantum states by means of differential geometry. A new approach is followed with the intention to facilitate the introduction of a more general theory in subsequent work. To this purpose, the emphasis is shifted from a manifold of strictly positive density matrices to a manifold of faithful quantum states on the C*-algebra of bounded linear operators. In addition, ideas from the parameter-free approach to information geometry are adopted. The underlying Hilbert space is assumed to be finite-dimensional. In this way, technicalities are avoided so that strong results are obtained, which one can hope to prove later on in a more general context. Two different atlases are introduced, one in which it is straightforward to show that the quantum states form a Banach manifold, the other which is compatible with the inner product of Bogoliubov and which yields affine coordinates for the exponential connection.
Quantum connections are defined by parallel transport operators acting on a Hilbert space. They transport tangent operators along paths in parameter space. The metric tensor of a Riemannian manifold is replaced by an inner product of pairs of operator fields, similar to the inner product of the Kubo-Mori formalism of Linear Response Theory. The metric is used to define the dual of a quantum connection.The gradient of the parallel transport operators is the quantum vector potential. It defines the covariant derivative of operator fields. The covariant derivatives are used to quantify the holonomy of the quantum connection. It is shown that a quantum connection is holonomic if and only if its dual is holonomic. If the parallel transport operators are unitary then an alpha-family of quantum connections can be defined in a way similar to Amari's alpha family of connections in Information Geometry. The minus alpha connection is the dual of the alpha connection. In particular, the alpha equal zero connection is self-dual. An operator field can be combined with a path in parameter space to produce a path in operator space. A definition is given for such a path in operator space to be autoparallel. The path in parameter space is then a geodesic for the induced connection.
Let (M,g) be a compact, connected and oriented Riemannian manifold. We denote D the space of smooth probability density functions on M. In this paper, we show that the Frechet manifold D is equipped with a Riemannian metric g^{D} and an affine connection \nabla^{D} which are infinite dimensional analogues of the Fisher metric and exponential connection in the context of information geometry. More precisely, we use Dombrowski's construction together with the couple (g^{D},\nabla^{D}) to get a (non-integrable) almost Hermitian structure on D, and we show that the corresponding fundamental 2-form is a symplectic form from which it is possible to recover the usual Schrodinger equation for a quantum particle living in M. These results echo a recent paper of the author where it is stressed that the Fisher metric and exponential connection are related (via Dombrowski's construction) to Kahler geometry and quantum mechanics in finite dimension.
Information geometry usually refers to a class of geometries characterized from metrics and corresponding connections related to the notion of statistical information during the process of parameter estimation on classical or quantum systems. We are aware of the fact that in parameter estimation we can't achieve perfectness. There is a lower bound, which is not zero, for the variance - covariance of the estimated parameter compared to the real parameter, the Cramer - Rao bound. In classical parameter estimation, this bound is given by the reciprocal of Fisher information and is unique in the sense that the Fisher is the unique monotone metric under stochastic transformations. In quantum parameter estimation there is a class of monotone metrics and the lowest possible bound, which determines the best quality of estimation, is given by that of the Symmetric Logarithmic Derivative (SLD). The term symmetric determines the derivative type, since there are more than one types due to the non-commutativity of operators, in general, in quantum mechanics. In our first published paper in the framework of this Ph.D. thesis, we are working beyond the class on monotone metrics. We are using the phase space to define a new statistical metric. This is achieved by introducing logarithmic derivatives through the Husimi Q distribution and consequently the new metric. As it turns out the related bound of Cramer - Rao type, can be greater even of this of the Symmetric Logarithmic Derivative. In our second paper, we refer to a particular bi-parameter family of generalized entropies used for the classification of complex systems. By defining a divergence for the discrete distribution, we consequently define a metric and the corresponding Levi - Civita connection, which depend on the two parameters say c and d. So, we are having geometrical objects with statistical content like the Cramer - Rao bound and the scalar curvature, which is characteristic (and different) for every class of complex systems. In our third paper, we are treating the above parameters c and d as coordinates of a two dimensional manifold. By calculating the Fisher metric and the corresponding scalar curvature we observe particular values of c and d for which we are making the conjecture that they related with particular properties of complex systems. Finally, we are highlighting a correlation of the curvature of a particular manifold and quantum entanglement. In particular, we are using the entropy of entanglement to define a two dimensional manifold, which we embed into three dimensional real space. We find that in the region which this surface has maximum curvature we have product quantum states and only there.
The manifold under consideration consists of the faithful normal states on a sigma-finite von Neumann algebra in standard form. Tangent planes and approximate tangent planes are discussed. A relative entropy/divergence function is assumed to be given. It is used to generalize the notion of an exponential arc connecting one state to another. The generator of the exponential arc is shown to be unique up to an additive constant. In the case of Araki’s relative entropy, every self-adjoint element of the von Neumann algebra generates an exponential arc. The generators of the composed exponential arcs are shown to add up. The metric derived from Araki’s relative entropy is shown to reproduce the Kubo–Mori metric. The latter is the metric used in linear response theory. The e- and m-connections describe a dual pair of geometries. Any finite number of linearly independent generators determines a submanifold of states connected to a given reference state by an exponential arc. Such a submanifold is a quantum generalization of a dually flat statistical manifold.