  When an open system is contacted with several thermal baths, the entropy
produced by the irreversible processes
($dS_{\mathrm{i}}=dS-\sum_{\alpha}\text{\dj}Q_{\alpha}/T_{\alpha}$) keeps
increasing, and this entropy production rate is always non-negative. But when
the system is contacted with some non-thermal baths containing quantum
coherence or squeezing, this entropy production formula does not apply. In this
paper, we study the increasing rate of the mutual information between the open
system and its environment. When the baths are canonical thermal ones, we prove
that this mutual information production rate could exactly return to the
previous entropy production rate. Further, we study an example of a single
boson mode contacted with multiple squeezed thermal baths, where the
conventional entropy production rate does not apply, and we find that this
mutual information production rate still keeps non-negative, which means the
monotonic increasing of the correlation between the system and its environment.
