  The classical Gibbs paradox concerns the entropy change upon mixing two
gases. Whether an observer assigns an entropy increase to the process depends
on their ability to distinguish the gases. A resolution is that an "ignorant"
observer, who cannot distinguish the gases, has no way of extracting work by
mixing them. Moving the thought experiment into the quantum realm, we reveal
new and surprising behaviour: the ignorant observer can extract work from
mixing different gases, even if the gases cannot be directly distinguished.
Moreover, in the macroscopic limit, the quantum case diverges from the
classical ideal gas: as much work can be extracted as if the gases were fully
distinguishable. We show that the ignorant observer assigns more microstates to
the system than found by naive counting in semiclassical statistical mechanics.
This demonstrates the importance of accounting for the level of knowledge of an
observer, and its implications for genuinely quantum modifications to
thermodynamics.
