Inspired by the second law of thermodynamics, we study the change in subsystem entropy generated by dynamical unitary evolution of a product state in a bipartite system. Working at leading order in perturbative interactions, we prove that the quantum n-Tsallis entropy of a subsystem never decreases, provided that subsystem is initialized as a statistical mixture of states of equal probability. This is true for any choice of interactions and any initialization of the complementary subsystem. When this condition on the initial state is violated, it is always possible to explicitly construct a "Maxwell's demon'' process that decreases the subsystem entropy. Remarkably, for the case of particle scattering, the circuit diagrams corresponding to n-Tsallis entropy are the same as the on-shell diagrams that have appeared in the modern scattering amplitudes program, and the entropy growth is intimately related to the nonnegativity of cross-sections.