Helstrom geometry with time-dependent spectral calibration

Abstract

In a recent work we introduced a finite scalar-qubit-scalar spectral triple whose Connes distance reproduces the Bloch-ball trace metric up to a factor $2/Λ$, where $Λ^2=μ_L^2+μ_R^2$ for isotropic link strengths $μ_L,μ_R$ appearing in the proposed Dirac operator. Here we analyse time-dependent variation of these links, independent of the qubit dynamics. In doing so, we separate the distinguishability and calibration contributions to the spectral rate. A depolarizing example demonstrates spectral growth despite a monotonic contraction of the trace-distance. More generally, we characterize the admissible trace-distance histories compatible with a given positive spectral record and exhibit contracting and reviving realizations. At fixed $Λ$, we also study a sinusoidal link imbalance that preserves the Dirac spectrum and qubit metric, while the scalar-scalar distance oscillates with half the period of the imbalance.