  Quantum direct coding or Schumacher compression generalised the ideas of
Shannon theory, gave an operational meaning to the von Neumann entropy and
established the term qubit. But remembering that information processing is
carried out by physical processes prompts one to wonder what thermodynamic
resources are required to compress quantum information and how they constrain
one's ability to perform this task. That is, if Alice and Bob only have access
to thermal quantum states and clocks with finite accuracy, how well can they
measure, encode and decode pure quantum state messages? In this work we examine
these questions by modelling Alice's typical measurement as a unitary involving
a measurement probe, investigating imperfect timekeeping on encoding and
decoding and considering the role of temperature in Bob's appended qubits. In
doing so, we derive fidelity bounds for this protocol involving the
correlations Alice can form with their measurement probe, the variance of the
clock's ticks and the temperature of Bob's qubits. Finally, we give an insight
into the entropy produced by these two agents throughout the compression
protocol by relating the resources they use to a quantum thermodynamic cooling
protocol.
