We derive an equality for non-equilibrium statistical mechanics in finite-dimensional quantum
systems. The equality concerns the worst-case work output of a time-dependent Hamiltonian pro-
tocol in the presence of a Markovian heat bath. It has has the form“worst-case work = penalty -
optimum” The equality holds for all rates of changing the Hamiltonian and can be used to derive
the optimum by setting the penalty to 0. The optimum term contains the max entropy of the
initial state, rather than the von Neumann entropy, thus recovering recent results from single-shot
statistical mechanics. Energy coherences can arise during the protocol but are assumed not to be
present initially. We apply the equality to an electron box.