  Landauer's principle provides a link between Shannon's information entropy
and Clausius' thermodynamical entropy. We set up here a basic formula for the
incremental free energy of a quantum channel, possibly relative to infinite
systems, naturally arising by an Operator Algebraic point of view. By the
Tomita-Takesaki modular theory, we can indeed describe a canonical evolution
associated with a quantum channel state transfer. Such evolution is implemented
both by a modular Hamiltonian and a physical Hamiltonian, the latter being
determined by its functoriality properties. This allows us to make an intrinsic
analysis, extending our QFT index formula, but without any a priori given
dynamics; the associated incremental free energy is related to the logarithm of
the Jones index and is thus quantised. This leads to a general lower bound for
the incremental free energy of an irreversible quantum channel which is half of
the Landauer bound, and to further bounds corresponding to the discrete series
of the Jones index. In the finite dimensional context, or in the case of DHR
charges in QFT, where the dimension is a positive integer, our lower bound
agrees with Landauer's bound.
