Abstract

We derive the reduced-spin von Neumann entropy generated by unitary Stern–Gerlach spin–position entanglement in a closed system. For any pure two-branch state with normalized spatial branches, the reduced-spin spectrum and entropy are determined entirely by the initial spin population p and the magnitude |γ| of the spatial branch overlap. This constitutes an exact, model-independent two-branch overlap law. For equal-width, unchirped Gaussian branches, |γ| = exp(−κ), where κ = Δx2/(8σx2) + σx2Δp2/(2ℏ2) is a quadratic phase-space distinguishability parameter that combines relative position and momentum displacements. The Gaussian entropy can therefore be written as Ss(κ,p), although this particular phase-space expression for κ does not generally apply to arbitrary wave-packet shapes. We derive the reduced density matrix and its spectrum, the equal-width Gaussian reduction, the constant-gradient form of κ(t), the weak-distinguishability onset and saturation limits, and an entropy response function. We also derive the overlap for freely spreading, chirped equal-width Gaussians and for unequal-width unchirped Gaussians, for which the fixed-width parameter κ is replaced by the corresponding Gaussian overlap coordinate. Different apparatus settings generally produce different entropy trajectories in time; at a fixed input population, these trajectories collapse only after reparametrization in terms of the appropriate overlap coordinate.

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