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Infinite Dimensional Analysis Quantum Probability and Related Topics

DUAL GEOMETRY OF THE WIGNER–YANASE–DYSON INFORMATION CONTENT

Hiroshi Hasegawa

Published Sep 1, 2003• 19 citations
Publisher

Abstract

We develop a non-parametric information geometry on finite-dimensional matrix manifolds by using the Fréchet differentiation. Taking the simplest prototype Riemannian metric form [Formula: see text], [Formula: see text] with the Fréchet derivative D on a pair of smooth functions g(ρ) and g*(ρ) of the density matrix ρ, we prove the WYD identification theorem: this metric is identified with the normalized Wigner–Yanase–Dyson skew information (the WYD metric), if and only if the metric form satisfies the monotonicity under every stochastic mapping T on ρ and A: [Formula: see text]. On this basis, we establish (a) a fine structure of the partial order in the set ℱ of all monotone metrics such that ℱ power ∪ℱ WYD forms a linearly ordered subset of ℱ with the same mini-max bound, where ℱ power (the power-mean metrics) interpolates the Bures and the WYD metrics (b) a characterization of the quasi-entropy S (ρ, σ)= Tr F (Δσ, ρ)ρ induced by the metric-characterizing function f WYD (x) of the WYD metric (c) an affine connection on the above metric which is torsionless to guarantee the quantum version of the ±α-connection, provided α∈[-3, 3].

DOI
10.1142/s021902570300133x

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A CHARACTERISATION OF WIGNER–YANASE SKEW INFORMATION AMONG STATISTICALLY MONOTONE METRICS

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Let [Formula: see text] be the space of n × n complex matrices endowed with the Hilbert–Schmidt scalar product, let Sn be the unit sphere of Mn and let Dn⊂ Mn be the space of strictly positive density matrices. We show that the scalar product over Dn introduced by Gibilisco and Isola3 (that is the scalar product induced by the map [Formula: see text]) coincides with the Wigner–Yanase monotone metric.

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The geometrical formulation of quantum statistics and the theory of measures of quantum information are intimately connected by the introduction of the notion of metric adjusted skew information.We survey the area with a focus on the ‘representing operator monotone functions’. In particular, we exhibit a new order structure that renders the set of metric adjusted skew informations into a lattice with a maximal element. The Wigner–Yanase–Dyson skew informations (with parameter p) are increasing with respect to this order structure for 0 < p ≤ 1/2 and decreasing for 1/2 ≤ p < 1 with maximum in the Wigner–Yanase skew information. Introduction The geometrical formulation of quantum statistics and the theory of measures of quantum information are two distinct theories with separate motivations and histories, and for a long time they did not seem to be related in any way. Early contributions which view quantum information in statistical terms were given by Hasegawa and Petz (Hasegawa and Petz 1996), and by Luo (Luo 2003a, Luo 2003b, Luo 2005). Today the two theories are largely fused with the notion of metric (or metrically) adjusted skew information and a common set of tools and techniques involving certain operator monotone functions and their representations. The geometrical approach to statistics reveals its fundamental nature when we try to generalise classical statistics to the quantum setting. The key to obtaining quantisation of the Fisher information is to consider it as a Riemannian metric with a certain behaviour in the presence of noise.

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Fisher Information Metrics and Entropic Dynamics in Quantum Statistical Manifolds: A Motivic Approach

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Mathematical Applications of Science Fiction We present a rigorous derivation of the Quantum FisherInformation Metric (QFIM) within the framework of derivedalgebraic geometry and non-commutative spectral theory.By treating the manifold of quantum states as a modulispace of stable sheaves over a non-commutative C∗-algebra,we establish a canonical isomorphism between the tangentbundle of the statistical manifold and the first derived coho-mology group of the entropic structure sheaf. We introducethe Malcor-Bures Spectral Sequence, which converges to theBures metric, and prove that the entropic dynamics gener-ated by the gradient flow of the von Neumann entropy isgeodesically complete with respect to the Wigner-Yanase-Dyson skew information metric. Furthermore, we analyze thestability of these dynamics using motivic L-functions associ-ated with the modular automorphism group, demonstratingthat the information geometry admits a unique hyper-Kählerstructure in the thermodynamic limit.

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Εφαρμογές της γεωμετρίας της πληροφορίας

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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

Non-hermitian extension of uncertainty relation

Kenjiro Yanagi, 研二郎 柳

which is known as the Wigner-Yanase-Dyson skew information. Recently it is shown that these skew informations are connected to special choices of quantum Fisher information in [3]. The family of all quantum Fisher informations is parametrized by a certain class of operator monotone functions Fop which were justified in [9]. The Wigner-Yanase skew information and Wigner-Yanase-Dyson skew information are given by the following operator monotone functions

数理解析研究所講究録•Jul 1, 2014•0 citations
We develop a non-parametric information geometry on finite-dimensional matrix manifolds by using the Fréchet differentiation. Taking the simplest prototype Riemannian metric form [Formula: see text], [Formula: see text] with the Fréchet derivative D on a pair of smooth functions g(ρ) and g*(ρ) of the density matrix ρ, we prove the WYD identification theorem: this metric is identified with the normalized Wigner–Yanase–Dyson skew information (the WYD metric), if and only if the metric form satisfies the monotonicity under every stochastic mapping T on ρ and A: [Formula: see text]. On this basis, we establish (a) a fine structure of the partial order in the set ℱ of all monotone metrics such that ℱ power ∪ℱ WYD forms a linearly ordered subset of ℱ with the same mini-max bound, where ℱ power (the power-mean metrics) interpolates the Bures and the WYD metrics (b) a characterization of the quasi-entropy S (ρ, σ)= Tr F (Δσ, ρ)ρ induced by the metric-characterizing function f WYD (x) of the WYD metric (c) an affine connection on the above metric which is torsionless to guarantee the quantum version of the ±α-connection, provided α∈[-3, 3].
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.
The geometrical formulation of quantum statistics and the theory of measures of quantum information are intimately connected by the introduction of the notion of metric adjusted skew information.We survey the area with a focus on the ‘representing operator monotone functions’. In particular, we exhibit a new order structure that renders the set of metric adjusted skew informations into a lattice with a maximal element. The Wigner–Yanase–Dyson skew informations (with parameter p) are increasing with respect to this order structure for 0 < p ≤ 1/2 and decreasing for 1/2 ≤ p < 1 with maximum in the Wigner–Yanase skew information. Introduction The geometrical formulation of quantum statistics and the theory of measures of quantum information are two distinct theories with separate motivations and histories, and for a long time they did not seem to be related in any way. Early contributions which view quantum information in statistical terms were given by Hasegawa and Petz (Hasegawa and Petz 1996), and by Luo (Luo 2003a, Luo 2003b, Luo 2005). Today the two theories are largely fused with the notion of metric (or metrically) adjusted skew information and a common set of tools and techniques involving certain operator monotone functions and their representations. The geometrical approach to statistics reveals its fundamental nature when we try to generalise classical statistics to the quantum setting. The key to obtaining quantisation of the Fisher information is to consider it as a Riemannian metric with a certain behaviour in the presence of noise.
Mathematical Applications of Science Fiction We present a rigorous derivation of the Quantum FisherInformation Metric (QFIM) within the framework of derivedalgebraic geometry and non-commutative spectral theory.By treating the manifold of quantum states as a modulispace of stable sheaves over a non-commutative C∗-algebra,we establish a canonical isomorphism between the tangentbundle of the statistical manifold and the first derived coho-mology group of the entropic structure sheaf. We introducethe Malcor-Bures Spectral Sequence, which converges to theBures metric, and prove that the entropic dynamics gener-ated by the gradient flow of the von Neumann entropy isgeodesically complete with respect to the Wigner-Yanase-Dyson skew information metric. Furthermore, we analyze thestability of these dynamics using motivic L-functions associ-ated with the modular automorphism group, demonstratingthat the information geometry admits a unique hyper-Kählerstructure in the thermodynamic limit.
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.