On the Curvature of Monotone Metrics and a Conjecture Concerning the Kubo-Mori Metric

Monotone Riemannian metrics and dynamic structure factor in condensed matter physics

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, Chencov 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 dyn…

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

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 cla…

Silver mean conjectures for 15-d volumes and 14-d hyperareas of the separable two-qubit systems

Extensive numerical integration results lead us to conjecture that the silver mean, that is, s = \sqrt{2}-1 = .414214 plays a fundamental role in certain geometries (those given by monotone metrics) imposable on the 15-dimensional convex set of two-qubit systems. For example, we hypothesize that the volume of separable two-qubit states, as measured in terms of (four times) the minimal monotone or Bures metric is s/3, and 10s in terms of (four times) the Kubo-Mori monotone metric. Also, we conjecture, in terms of (four times) the Bures metric, that that part of the 14-dimensional boundary of se…

高斯量子态的曲率

Curvature of Gaussian quantum states

本文研究高斯量子态流形的几何曲率，发现曲率随冯·诺依曼熵严格单调，可作为态不确定性度量，并为连续变量系统中的佩茨猜想提供证据。