Operational meaning of the classical fidelity and the path length in Fisher-Kubo-Mori-Bogoliubov geometry

TL;DR

The paper tackles the problem of giving an operational meaning to global statistical distances in Fisher--KMB geometry by linking path length to entropy production in near-reversible quantum state transport. Using a reservoir-based microscopic model, the authors show that the minimum entropy production along a path $\gamma$ is $\Delta S_\gamma = \frac{\ell_\gamma^2}{2N}$ and, in the classical limit with geodesic length $\ell_\gamma(p,q)=2\arccos F(p,q)$, the bound becomes $\Delta S = \frac{2}{N}(\arccos F)^2$. Introducing the step density $\nu = N/\ell_\gamma$, they obtain $\Delta S_\gamma(\ell) = \frac{\ell}{2\nu}$, which gives an operational meaning to the Fisher--KMB path length as a per-unit-length entropy cost in near-reversible transport. The work connects to broader geometric frameworks (Kantorovich--Wasserstein, Weinhold--Ruppeiner) and provides the first operational interpretation of the global Fisher distance and the classical Bhattacharyya fidelity, with future directions toward continuous protocols.

Abstract

We show that the minimum entropy production in near-reversible quantum state transport along a path is simple function of the path length measured according to the Fisher-KMB metrics. Hence the sharp values of path lengths, also called statistical lengths, obtain operational meaning to quantify the residual irreversibility in near-reversible state transport. In the classical limit, the Bhattacharyya fidelity obtains a sharp operational meaning after eighty years.