Operational Meaning of Classical Fidelity and Path Length in Kubo-Mori-Bogoliubov Fisher Geometry.

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