Spectral-Gap Bounds and Timescales for Purity Loss in Hamiltonian--Pointer Interactions

Abstract

We investigate the purity dynamics of a quantum system coupled to a continuous-variable pointer through a von Neumann-type interaction Hamiltonian of the form $\hat H_{\mathrm{int}}=g\,\hat H\otimes\hat p$. For an initially Gaussian pointer state, the interaction generates energy-dependent conditional translations whose mutual overlaps are determined explicitly by the populated spectral separations of the system Hamiltonian. After tracing out the pointer degrees of freedom, we obtain the reduced density operator and derive an exact analytical expression for the time-dependent purity. Using this expression, we establish two-sided purity bounds governed by the minimum and maximum nonzero energy gaps on the populated spectral support. These bounds provide a state-dependent spectral characterization of the loss of purity and become exact for two-level systems. We further show that the short-time decrease of purity is controlled by the Hamiltonian variance of the initial state, with $\mathcal P''(0)=-(g^2/σ^2) \operatorname{Var}_{ψ_S}(\hat H)$. In addition, an explicit sufficient timescale is derived for the purity to approach its asymptotic value within a prescribed tolerance, revealing the scaling $t_{\varepsilon}\proptoσ/(|g|Δ_{\min})$. Finally, the general results are illustrated for an equally weighted $N$-level system with an equally spaced spectrum, for which the asymptotic purity is $1/N$. The analysis clarifies the distinct roles of spectral separation, energy variance, coupling strength, and pointer width in Hamiltonian-conditioned purity loss.