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.

The paper asks whether increasing Connes spectral distance witnesses quantum information backflow when the Dirac operator defining the metric varies independently of the qubit dynamics. It studies a finite scalar–qubit–scalar spectral triple with positive isotropic links $\mu_L(t),\mu_R(t)$. Building on the previously established identity $d(t)=2T(t)/\Lambda(t)$, where $T$ is trace distance and $\Lambda^2=\mu_L^2+\mu_R^2$, it separates distinguishability changes from calibration changes through $\dot d/d=\dot T/T-\dot\Lambda/\Lambda$. Explicit analytic constructions show that depolarizing evolution with $T(t)=T_0e^{-\gamma t}$ can exhibit spectral growth when the calibration scale decreases sufficiently quickly. More generally, for any positive differentiable spectral record on a finite interval, a differentiable trace-distance history is realizable by a common CPTP family and positive links if and only if $0<T(t)\leq T(0)\leq1$. Sufficiency follows from a random-unitary dephasing construction. Consequently, the same spectral record can describe either monotonic contraction or trace-distance revival. Simultaneous scalar-reference distances recover the missing calibration scale and hence $T(t)$. At fixed $\Lambda$, varying the link imbalance $\beta=\log(\mu_R/\mu_L)$ preserves both the Dirac spectrum and the qubit metric, while the scalar–scalar distance becomes $2\cosh\beta/\Lambda$. A sinusoidal imbalance therefore produces a scalar-distance oscillation at twice its frequency. The analysis assumes isotropic identity links and known spectral distances. Sampling-error propagation, physical implementation, and anisotropic links are left open; anisotropy need not admit the same scalar calibration.