  We investigate a weak measurement described by a von Neumann type interaction
$\hat{A}\otimes \hat{p}^2$, where $\hat{A}$ is a system observable and
$\hat{p}^2$ is a measurement pointer observable. We consider the weak
measurement in terms of thermodynamics by adopting a mixed Gaussian state as a
quantum state of the measurement pointer. We show that Maxwell's demon appears
as a measure who carries out post-selections. It is found that, even if the
demon only knows the weak value, a difference in the von Neumann entropy
between the initial and final system can be the QC mutual information contents,
which is the maximum amount of obtainable information by a measurement.
Besides, our study indicates that a temperature of the system described by this
interaction is amplified by weak value amplification. In addition, we show that
this demon can be realized in an atomic system.
