Reexamination of pure qubit work extraction

Many work extraction or information erasure processes in the literature involve the raising and lowering of energy levels via external fields. But even if the actual system is treated quantum mechanically, the field is assumed to be classical and of infinite strength, hence not developing any correlations with the system or experiencing back-actions. We extend these considerations to a fully quantum mechanical treatment by studying a spin-1/2 particle coupled to a finite-sized directional quantum reference frame, a spin-l system, which models an external field. With this concrete model together with a bosonic thermal bath, we analyze the back-action a finite-size field suffers during a quantum-mechanical work extraction process and the effect this has on the extractable work and highlight a range of assumptions commonly made when considering such processes. The well-known semiclassical treatment of work extraction from a pure qubit predicts a maximum extractable work W = kT log 2 for a quasistatic process, which holds as a strict upper bound in the fully quantum mechanical case and is attained only in the classical limit. We also address the problem of emergent local time dependence in a joint system with a globally fixed Hamiltonian.