Hamiltonian Composite Dynamics Can Almost Always Lead To Negative Reduced Dynamics

Complete positivity is a ubiquitous assumption in the study of quantum systems interacting with the environment, despite repeated efforts to point out that the assumption is not empirically justified. It will be shown that Hamiltonian evolution of a quantum system and its environment can be negative (i.e.\ not completely positive) in the energy basis, by showing that such evolution is {\it almost always} negative for given initial conditions. Ignoring or "correcting" experimental data that is not completely positive may cause the loss of important information regarding system-environment correlations and coupling. A relationship between the negativity of an evolution and the eigenvalues of the Hamiltonian will be shown, and experimental verification of negative reduced dynamics will be proposed.