  We discuss the local equilibration of closed systems using the relative
purity, a paradigmatic information-theoretic distinguishability measure that
finds applications ranging from quantum metrology to quantum speed limits.
First we obtain an upper bound on the average size of the fluctuations on the
relative purity: it depends on the effective dimension resembling the bound
obtained with the trace distance. Second, we investigate the dynamics of
relative purity and its rate of change as a probe of the speed of fluctuations
around the equilibrium. In turn, such speed captures the notion of how fast
some nonequilibrium state approaches the steady state under the local
nonunitary dynamics, somehow giving the information of the quantum speed limit
towards the equilibration. We show that the size of fluctuations depends on the
quantum coherences of the initial state with respect to the eigenbasis of the
Hamiltonian, also addressing the role played by the correlations between system
and reservoir into the averaged speed. Finally, we have derived a family of
lower bounds on the time of evolution between these states, thus obtaining an
estimate for the equilibration time at the local level. These results could be
of interest to the subjects of equilibration, quantum speed limits, and also
quantum metrology.
