Information geometry of density matrices and state estimation

TL;DR

This paper explores the geometric structure of density matrices using information geometry, deriving a Riemannian metric and applying it to quantum state estimation, including new uncertainty relation corrections for mixed states.

Contribution

It introduces a unitary invariant Riemannian metric on density matrices derived from information geometry and applies it to quantum state estimation, providing higher-order uncertainty corrections.

Findings

Derived a unitary invariant Riemannian metric on density matrices.

Provided an alternative derivation using square-root density matrices.

Developed higher-order corrections to quantum uncertainty relations.

Abstract

Given a pure state vector |x> and a density matrix rho, the function p(x|rho)=<x|rho|x> defines a probability density on the space of pure states parameterised by density matrices. The associated Fisher-Rao information measure is used to define a unitary invariant Riemannian metric on the space of density matrices. An alternative derivation of the metric, based on square-root density matrices and trace norms, is provided. This is applied to the problem of quantum-state estimation. In the simplest case of unitary parameter estimation, new higher-order corrections to the uncertainty relations, applicable to general mixed states, are derived.