An Entropy–Time Law for Quantum Distinguishability and Quantum Search Dynamics

We derive an algorithm-independent entropy–time law governing quantum distinguishability.Starting from a general quantum state and a target hypothesis represented by a pure state,we introduce an intrinsic invariant S_sr defined by the inverse initial overlap. We prove that achieving constant statistical distinguishability requires time scaling as√S_sr, with a sharp prefactor π/4. The bound is saturated by optimal two-level dynamics,extends to multiple marked states, and remains valid under open-system perturbations. Numerical simulations confirm the theoretical scaling and demonstrate universal collapseunder rescaling by √S_sr. Standard quantum search bounds arise as a special case.