  Weak invariants are time-dependent observables with conserved expectation
values. Their fluctuations, however, do not remain constant in time. On the
assumption that time evolution of the state of an open quantum system is given
in terms of a completely positive map, the fluctuations monotonically grow even
if the map is not unital, in contrast to the fact that monotonic increases of
both the von Neumann entropy and R\'enyi entropy require the map to be unital.
In this way, the weak invariants describe temporal asymmetry in a manner
different from the entropies. A formula is presented for time evolution of the
covariance matrix associated with the weak invariants in the case when the
system density matrix obeys the Gorini-Kossakowski-Lindblad-Sudarshan equation.
