Characterize non-uniform priors for other quantum channels

Determine the reversibility, explicit spectrum or spectral gap, and decay rate governing how non-uniform stationary priors are forgotten for quantum channels beyond the depolarizing channel, particularly channels with smooth stationary densities whose density matrices relax to states other than the maximally mixed state.

Background

The paper derives a reverse stochastic differential equation for quantum state diffusion and analyzes the depolarizing channel in detail. For that channel, the pure-state diffusion is Brownian motion on complex projective space, the Fubini–Study measure is a uniform stationary prior, and the generator has an explicitly known spectrum and spectral gap. The discussion proposes extending this analysis to other channels with smooth, non-uniform stationary densities and notes that the relevant dynamical and convergence properties have not yet been established.

The unresolved aspects are the reversibility of the corresponding diffusion, its spectral data—at least its spectral gap—and the rate at which the reverse process forgets the chosen stationary prior. These quantities would determine whether analogous score-based generative constructions and error bounds can be developed beyond the depolarizing setting.