  We address a new setting where the second law is under question:
thermalizations in a quantum superposition of causal orders, enacted by the
so-called quantum switch. This superposition has been shown to be associated
with an increase in the communication capacity of the channels, yielding an
apparent violation of the data-processing inequality and a possibility to
separate hot from cold. We analyze the thermodynamics of this information
capacity increasing process. We show how the information capacity increase is
compatible with thermodynamics. We show that there may indeed be an information
capacity increase for consecutive thermalizations obeying the first and second
laws of thermodynamics if these are placed in an indefinite order and moreover
that only a significantly bounded increase is possible. The increase comes at
the cost of consuming a thermodynamic resource, the free energy of coherence
associated with the switch.
