Axiomatic Foundation of Quantum-Inspired Distance Metrics

TL;DR

This paper establishes an axiomatic foundation for quantum-inspired distance metrics, unifying and generalizing existing measures, and revealing their fundamental relationships and operational meanings in quantum information theory.

Contribution

It introduces a comprehensive axiomatic framework that characterizes quantum distances, proving the uniqueness of the Fubini-Study metric and connecting various distances through a hierarchy.

Findings

Fubini-Study metric is the unique geodesic distance under the axioms

Established relationships among different quantum distances

Provided operational interpretations linked to quantum tasks

Abstract

We develop a comprehensive axiomatic framework for quantum-inspired distance metrics on projective Hilbert spaces, providing a unified foundation that organizes and generalizes existing measures in quantum information theory. Starting from five fundamental axioms, projective invariance, unitary covariance, superposition sensitivity, entanglement awareness, and measurement contextuality, we show that any admissible distance depends solely on state overlap and establish the uniqueness of the Fubini-Study metric as the canonical geodesic distance. Our framework further yields a hierarchy of comparison results relating the Fubini-Study metric, Bures distance, Euclidean distance, measurement-based pseudometrics, and entanglement-sensitive distances. Key contributions include an entanglement-geometry complementarity principle, high-dimensional concentration bounds, and operational interpretations connecting distances to state discrimination and quantum metrology. This work places the geometry of quantum state spaces on a rigorous axiomatic footing, bridging abstract metric theory, information geometry, and operational quantum principles.