  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,
$\Delta S_n \geq 0$, 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,
$\Delta S_n < 0$. 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
$\Delta S_n \geq 0$ is intimately related to the nonnegativity of
cross-sections.
