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Quantum nonequilibrium equalities with absolute irreversibility

We derive quantum nonequilibrium equalities in absolutely irreversible processes. Here by absolute irreversibility we mean that in the backward process the density matrix does not return to the subspace spanned by those eigenvectors that have nonzero weight in the initial density matrix. Since the initial state of a memory and the postmeasurement state of the system are usually restricted to a subspace, absolute irreversibility occurs during the measurement and feedback processes. An additional entropy produced in absolute irreversible processes needs to be taken into account to derive nonequilibrium equalities. We discuss a model of a feedback control on a qubit system to illustrate the obtained equalities. By introducing $N$ heat baths each composed of a qubit and letting them interact with the system, we show how the entropy reduction via feedback control can be converted into work. An explicit form of extractable work in the presence of absolute irreversibility is given.