Exponential arcs in manifolds of quantum states

TL;DR

This paper explores the geometric structure of quantum state manifolds using exponential arcs, relative entropy, and dual connections, revealing a quantum generalization of dually flat statistical manifolds with applications in quantum information geometry.

Contribution

It introduces a generalized notion of exponential arcs in quantum state manifolds and analyzes their properties, including generator uniqueness and metric relations, extending classical information geometry to quantum settings.

Findings

Exponential arcs are uniquely generated up to an additive constant.

Generators of composed exponential arcs add up.

The derived metric reproduces the Kubo-Mori metric used in Linear Response Theory.

Abstract

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 selfadjoint element of the von Neumann algebra generates an exponential arc. The generators of 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. Such a submanifold is a quantum generalization of a dually flat statistical manifold.