Remarks on pairwise comparisons, transition amplitudes, and qubit states

TL;DR

This paper explores the relationship between pairwise comparisons, quantum states, and geometric phases, providing a conceptual framework linking quantum kinematics with comparison data.

Contribution

It introduces a unified perspective connecting pairwise comparison data with quantum geometric phases and invariants, enriching the understanding of quantum state comparisons.

Findings

Phase data define a U(1)-valued reciprocal pairwise comparison structure.

Triangular defects relate to normalized Bargmann invariants and geometric phases.

Constraints from Gram matrices of rank at most two affect realizability of transition data.

Abstract

We discuss a pairwise-comparison viewpoint on finite families of qubit states. Starting from transition amplitudes between pure states, we distinguish three associated levels of comparison data: complex amplitudes, transition probabilities, and phase-valued pairwise comparisons. In the non-orthogonal case, the phase data define a \(U(1)\)-valued reciprocal pairwise comparison structure. We show that the corresponding triangular defects are naturally related to normalized Bargmann invariants and therefore to geometric phases. This gives a simple interpretation of inconsistency-type quantities in terms of quantum kinematics. We also comment on realizability constraints coming from Gram matrices of rank at most two, and on the passage from unitary phase data to more general transition data. The aim of the paper is mainly conceptual: to isolate a common language between pairwise comparisons and elementary quantum geometry.