  Irreversible information processing cannot be carried out without some
inevitable thermodynamical work cost. This fundamental restriction, known as
Landauer's principle, is increasingly relevant today, as the energy dissipation
of computing devices impedes the development of their performance. Here we
determine the minimal work required to carry out any logical process, for
instance a computation. It is given by the entropy of the discarded information
conditional to the output of the computation. Our formula takes precisely into
account the statistically fluctuating work requirement of the logical process.
It enables the explicit calculation of practical scenarios, such as
computational circuits or quantum measurements. On the conceptual level, our
result gives a precise and operationally justified connection between
thermodynamic and information entropy, and explains the emergence of the
entropy state function in macroscopic thermodynamics.
