Conditional action and quantum versions of Maxwell's demon

We propose a new way of looking at the quantum Maxwell's demon problem in terms of conditional action. A "conditional action" on a system is a unitary time evolution, selected according to the result of a previous measurement, which can reduce the entropy of the system. However, any conditional action can be realized by an (unconditional) unitary time evolution of a larger system and a subsequent L\"uders measurement, whereby the entropy of the entire system is either increased or remains constant. We give some examples that illustrate and confirm our proposal, including the erasure of $N$ qubits and the Szilard engine, thus relating the present approach to the Szilard principle and the Landauer principle that have been discussed as possible solutions of the Maxwell's demon problem.