  According to Landauer's principle, erasure of information is the only part of
a computation process that unavoidably involves energy dissipation. If done
reversibly, such an erasure generates the minimal heat of $k_BT\ln 2$ per
erased bit of information. The goal of this work is to discuss the actual
reversal of the optimal erasure which can serve as the basis for the Maxwell's
demon operating with ultimate thermodynamic efficiency as dictated by the
second law of thermodynamics. The demon extracts $k_BT\ln 2$ of heat from an
equilibrium reservoir at temperature $T$ per one bit of information obtained
about the measured system used by the demon. We have analyzed this Maxwell's
demon in the situation when it uses a general quantum system with a discrete
spectrum of energy levels as its working body. In the case of the effectively
two-level system, which has been realized experimentally based on tunneling of
individual electron in a single-electron box [J.V. Koski et al., PNAS 111,
13786 (2014)], we also studied and minimized corrections to the ideal
reversible operation of the demon. These corrections include, in particular,
the non-adiabatic terms which are described by a version of the classical
fluctuation-dissipation theorem. The overall reversibility of the Maxwell's
demon requires, beside the reversibility of the intrinsic working body
dynamics, the reversibility of the measurement and feedback processes. The
single-electron demon can, in principle, be made fully reversible by developing
a thermodynamically reversible single-electron charge detector for measurements
of the individual charge states of the single-electron box.
