Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Axiomatic Foundation of Quantum-Inspired Distance Metrics

Maryam Bagherian

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.

Axiomatic Foundation of Quantum-Inspired Distance Metrics

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.
Paper Structure (36 sections, 27 theorems, 153 equations)

This paper contains 36 sections, 27 theorems, 153 equations.

Table of Contents

  1. Introduction
  2. Foundations of Quantum-Inspired Distance Metric Functions
  3. The Geometry of Pure Quantum States
  4. Structural Constraints on Quantum Distance Functions
  5. (i) Projective and Unitary Invariance.
  6. (ii) Operational Consistency.
  7. (iii) Contractivity Under Quantum Channels.
  8. A Hierarchy of Quantum Distance Functions
  9. (i) Geometric Distances
  10. (ii) Operational Distances
  11. (iii) Fidelity and Bures Distance
  12. (iv) Entanglement-Sensitive Distances
  13. (v) Computational Considerations
  14. Illustrative Examples
  15. Axiomatic Framework for Quantum-Inspired Distance Metric Functions
  16. ...and 21 more sections

Key Result

Theorem 3.9

Let $d$ be a mapping on $\mathcal{P}(\mathcal{H})$ satisfying Axiom ax:ray and Axiom ax:unitary_invariance. Then there exists a function such that for all nonzero $\psi,\varphi \in \mathcal{H}$,

Theorems & Definitions (84)

  • Definition 1.1: Separable and Entangled States
  • Definition 2.1: Hilbert Space
  • Definition 2.2: Projective Hilbert Space
  • Definition 2.3: Separable and Entangled States
  • Definition 2.4: Hilbert-Space Distance
  • Definition 2.5: Transition Probability
  • Definition 2.6: Fubini--Study metric
  • Definition 2.7: Trace Distance
  • Remark 2.8: Trace Distance for Pure States and Helstrom Bound
  • Definition 2.9: Fidelity
  • ...and 74 more