  Local memory - the ability to extract information from a subsystem about its
initial state - is a central feature of many-body localization. We introduce,
investigate, and compare several information-theoretic quantifications of
memory and discover a hierarchical relationship among them. We also find that
while the Holevo quantity is the most complete quantifier of memory, vastly
outperforming the imbalance, its decohered counterpart is significantly better
at capturing the critical properties of the many-body localization transition
at small system sizes. This motivates our suggestion that one can emulate the
thermodynamic limit by artifically decohering otherwise quantum quantities.
Applying this method to the von Neumann entropy results in critical exponents
consistent with analytic predictions, a feature missing from similar small
finite-size system treatments. In addition, the decohering process makes
experiments significantly simpler by avoiding quantum state tomography.
