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arXiv:2106.07599

On the relation between the monotone Riemannian metrics on the space of Gibbs thermal states and the linear response theory

N. S. Tonchev

Published Jun 14, 2021• 2 citations

Abstract

The proposed in J. Math. Phys. v.57,071903 (2016) analytical expansion of monotone (contractive) Riemannian metrics (called also quantum Fisher information(s)) in terms of moments of the dynamical structure factor (DSF) relative to an original intensive observable is reconsidered and extended. The new approach through the DSF which characterizes fully the set of monotone Riemannian metrics on the space of Gibbs thermal states is utilized to obtain an extension of the spectral presentation obtained for the Bogoliubov-Kubo-Mori metric (the generalized isothermal susceptibility) on the entire class of monotone Riemannian metrics. The obtained spectral presentation is the main point of our consideration. The last allows to present the one to one correspondence between monotone Riemannian metrics and operator monotone functions (which is a statement of the Petz theorem in the quantum information theory) in terms of the linear response theory. We show that monotone Riemannian metrics can be determined from the analysis of the infinite chain of equations of motion of the retarded Green's functions. Inequalities between the different metrics have been obtained as well. It is a demonstration that the analysis of information-theoretic problems has benefited from concepts of statistical mechanics and might cross-fertilize or extend both directions, and vice versa. We illustrate the presented approach on the calculation of the entire class of monotone (contractive) Riemannian metrics on the examples of some simple but instructive systems employed in various physical problems.

DOI: 10.48550/arxiv.2106.07599

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The proposed in J. Math. Phys. v.57,071903 (2016) analytical expansion of monotone (contractive) Riemannian metrics (called also quantum Fisher information(s)) in terms of moments of the dynamical structure factor (DSF) relative to an original intensive observable is reconsidered and extended. The new approach through the DSF which characterizes fully the set of monotone Riemannian metrics on the space of Gibbs thermal states is utilized to obtain an extension of the spectral presentation obtained for the Bogoliubov-Kubo-Mori metric (the generalized isothermal susceptibility) on the entire class of monotone Riemannian metrics. The obtained spectral presentation is the main point of our consideration. The last allows to present the one to one correspondence between monotone Riemannian metrics and operator monotone functions (which is a statement of the Petz theorem in the quantum information theory) in terms of the linear response theory. We show that monotone Riemannian metrics can be determined from the analysis of the infinite chain of equations of motion of the retarded Green's functions. Inequalities between the different metrics have been obtained as well. It is a demonstration that the analysis of information-theoretic problems has benefited from concepts of statistical mechanics and might cross-fertilize or extend both directions, and vice versa. We illustrate the presented approach on the calculation of the entire class of monotone (contractive) Riemannian metrics on the examples of some simple but instructive systems employed in various physical problems.

An analytical approach is developed to the problem of computation of monotone Riemannian metrics (e.g., Bogoliubov-Kubo-Mori, Bures, Chernoff, etc.) on the set of quantum states. The obtained expressions originate from the Morozova, C̆ encov, and Petz correspondence of monotone metrics to operator monotone functions. The used mathematical technique provides analytical expansions in terms of the thermodynamic mean values of iterated (nested) commutators of a model Hamiltonian T with the operator S involved through the control parameter h. Due to the sum rules for the frequency moments of the dynamic structure factor, new presentations for the monotone Riemannian metrics are obtained. Particularly, relations between any monotone Riemannian metric and the usual thermodynamic susceptibility or the variance of the operator S are discussed. If the symmetry properties of the Hamiltonian are given in terms of generators of some Lie algebra, the obtained expansions may be evaluated in a closed form. These issues are tested on a class of model systems studied in condensed matter physics.

A Riemannian metric is defined on the state space of a finite quantum system by the canonical correlation (or Kubo–Mori/Bogoliubov scalar product). This metric is infinitesimally induced by the (nonsymmetric) relative entropy functional or the von Neumann entropy of density matrices. Hence its geometry expresses maximal uncertainty. It is proven that the metric is monotone under stochastic mappings, however, an example shows that it is not the only such Riemannian metric. This fact is remarkable because in the probabilistic case, the Markovian monotonicity property characterizes the Fisher information metric. The essential difference appears in the curvatures of a classical state space and a quantum one. A conjecture is made that the scalar curvature is monotone with respect to the ‘‘more mixed’’ (statistical) partial order of density matrices. Furthermore, an information inequality resembling the Cramér–Rao inequality of classical statistics is established. The inequality provides a lower bound for the canonical correlation matrix (of an unbiased observable) and it is saturated when a (partial) observation level and the corresponding family of coarse-grained states are considered.

Morozova and Chentsov (Morozova and Chentsov 90) studied Riemannian metrics on the set of probability measures. They showed that, up to a constant factor, the Fisher information is the only Riemannian metric which is monotone under stochastic transformation. Sarovar and Milburn (Sarovar and Milburn 06) computed an upper bound on the Fisher information for one-parameter channels. In (O'Loan 07) we extended their bound to an upper bound on the Fisher information of multi-parameter families of states; we call this the SM quantum information. Petz and Sudár (Petz 95) characterized fully the set of monotone metrics on the space of all density matrices. We analyse the SM quantum information in light of their work. We show that the SM quantum information is not a well-defined metric on the space of density matrices: different choices of phase of the eigenvectors lead to different metrics. We define a new metric C_L as a lower bound among the SM quantum informations. We look at properties of C_L and show that it is invariant but not monotone.

The quantum analog of the Fisher information metric of a probability simplex is searched and several Riemannian metrics on the set of positive definite density matrices are studied. Some of them appeared in the literature in connection with Cramér–Rao-type inequalities or the generalization of the Berry phase to mixed states. They are shown to be stochastically monotone here. All stochastically monotone Riemannian metrics are characterized by means of operator monotone functions and it is proven that there exist a maximal and a minimal among them. A class of metrics can be extended to pure states and a constant multiple of the Fubini–Study metric appears in the extension.

Chentsov studied Riemannian metrics on the set of probability measures from the point of view of decision theory. He proved that up to a constant factor the Fisher information is the only metric which is monotone under stochastic transformations. The present paper deals with monotone metrics on the space of finite density matrices on the basis motivated by quantum mechanics. A characterization of those metrics is given in terms of operator monotone functions. Several concrete metrics are constructed and analyzed, in particular, instead of the uniqueness in the probabilistic case, there is a large class of monotone metrics. Some of those appeared already in the physics literature a long time ago. A limiting procedure to pure states is discussed as well.

The canonical correlation or Kubo-Mori scalar product on the state space of a finite quantum system is a natural generalization of the classical Fisher metric. This metric is induced by the von Neumann entropy or the relative entropy of the quantum mechanical states. An important conjecture of Petz that the scalar curvature of the state space with Kubo-Mori scalar product as Riemannian metric is monotone with respect to the majorisation relation of states: the scalar curvature is increases if one goes to more mixed states. We give an appropriate grouping for the summands in the expression for the scalar curvature. The conjecture will follows from the monotonicity of the summands. We prove the monotonicity for some of these summands and we give numerical evidences that the remaining terms are monotone too. Note that the real density matrices form a submanifold of the complex density matrices. We prove that if Petz's conjecture true for complex density matrices then it is true for real density matrices too.