Statistical speed of quantum states: Generalized quantum Fisher information and Schatten speed

TL;DR

This paper introduces a unified framework for quantum statistical speeds using generalized quantum Fisher information and Schatten norms, providing insights into measurement optimization and bounds for quantum evolution speed.

Contribution

It extends the concept of quantum statistical speed to a broader family of measures, including Schatten norms, and explores their properties, optimal measurements, and physical interpretations.

Findings

Unified framework for quantum statistical speeds

Derived bounds on the speed of separable states

Linked trace speed to quantum Cramér-Rao bound for phase estimation

Abstract

We analyze families of measures for the quantum statistical speed which include as special cases the quantum Fisher information, the trace speed, i.e., the quantum statistical speed obtained from the trace distance, and more general quantifiers obtained from the family of Schatten norms. These measures quantify the statistical speed under generic quantum evolutions and are obtained by maximizing classical measures over all possible quantum measurements. We discuss general properties, optimal measurements, and upper bounds on the speed of separable states. We further provide a physical interpretation for the trace speed by linking it to an analog of the quantum Cram\'{e}r-Rao bound for median-unbiased quantum phase estimation.