  The Second Law of Thermodynamics states that the entropy of a closed system
is non-decreasing. Discussing the Second Law in the quantum world poses new
challenges and provides new opportunities, involving fundamental
quantum-information-theoretic questions and novel quantum-engineered devices.
In quantum mechanics, systems with an evolution described by a so-called unital
quantum channel evolve with a non-decreasing entropy. Here, we seek the
opposite, a system described by a non-unital and, furthermore,
energy-conserving channel that describes a system whose entropy decreases with
time. We propose a setup involving a mesoscopic four-lead scatterer augmented
by a micro-environment in the form of a spin that realizes this goal. Within
this non-unital and energy-conserving quantum channel, the micro-environment
acts with two non-commuting operations on the system in an autonomous way. We
find, that the process corresponds to a partial exchange or swap between the
system and environment quantum states, with the system's entropy decreasing if
the environment's state is more pure. This entropy-decreasing process is
naturally expressed through the action of a quantum Maxwell demon and we
propose a quantum-thermodynamic engine with four qubits that extracts work from
a single heat reservoir when provided with a reservoir of pure qubits. The
special feature of this engine, which derives from the energy-conservation in
the non-unital quantum channel, is its separation into two cycles, a working
cycle and an entropy cycle, allowing to run this engine with no local waste
heat.
