Reconstructing thermal states using dimensionally limited probes : A Model for Limited Control & Memory in Quantum Thermodynamics

Whilst the complexity of acquiring knowledge of a quantum state has been extensively studied in the fields of quantum tomography and quantum learning, a physical understanding of its operational role and cost in quantum thermodynamics is lacking. Knowledge is central to thermodynamics, as exemplified by Maxwell's demon thought experiment, where a demonic agent is able to extract paradoxical amounts of work - reconciled by the thermodynamic costs of acquiring this knowledge. In this work, we address this gap by extending unitary models of measurement to incorporate the resources available to an agent. We view an agent's knowledge of a quantum state as their ability to reconstruct it unitarily given access to states with partial knowledge of the true state. In our model, an agent correlates an unknown d-dimensional system, with copies of a k-dimensional probe $kleq d$, which are then used to unitarily reconstruct an estimate state in d-dimensional memories. We find that this framework is a unitary representation of coarse-grained POVMs. As an application, we investigate the role of knowledge in an extended Szilard Engine scenario.