  Information processing, quantum or classical, relies on channels transforming
multiple input states to different corresponding outputs. Previous research has
established bounds on the thermodynamic resources required for such operations,
but no protocols have been specified for their optimal implementation. For the
insightful case of qubits, we here develop explicit protocols to transform two
states in an energetically optimal manner. We first prove conditions on the
feasibility of carrying out such transformations at all, and then quantify the
achievable work extraction. Our results uncover a fundamental incompatibility
between the thermodynamic ideal of slow, quasistatic processes and the
information-theoretic requirement to preserve distinguishablity between
different possible output states.
