  The minimal heat that must be released to the reservoir when a single input
$x$ is mapped to a single output $y$ is considered. It was previously argued by
Zurek in an informal style that the minimal heat should be $K(x|y)$. Recently,
Kolchinsky has rigorously proved a generalized Zurek's bound in the Hamiltonian
"two-point measurement" scheme, which implies a trade-off between heat, noise
and protocol complexity in a computation from $x$ to $y$. Here we propose
another proof of the generalized Zurek's bound from a detailed fluctuation
theorem, showing that the generalized Zurek's bound holds in any system that
satisfies a DFT. Several applications of the bound are also discussed.
