  In systems undergoing localization-delocalization quantum phase transitions
due to disorder or monitoring, there is a crucial need for robust methods
capable of distinguishing phases and uncovering their intrinsic properties. In
this work, we develop a process of finding a Slater determinant representation
of free-fermion wave functions that accurately characterizes localized
particles, a procedure we dub "unscrambling". The central idea is to minimize
the overlap between envelopes of single-particle wave functions or,
equivalently, to maximize the inverse participation ratio of each orbital. This
numerically efficient methodology can differentiate between distinct types of
wave functions: exponentially localized, power-law localized, and conformal
critical, also revealing the underlying physics of these states. The method is
readily extendable to systems in higher dimensions. Furthermore, we apply this
approach to a more challenging problem involving disordered monitored free
fermions in one dimension, where the unscrambling process unveils the presence
of a conformal critical phase and a localized area-law quantum Zeno phase.
Importantly, our method can also be extended to free fermion systems without
particle number conservation, which we demonstrate by estimating the phase
diagram of $\mathbb{Z}_2$-symmetric disordered monitored free fermions. Our
results unlock the potential of utilizing single-particle wave functions to
gain valuable insights into the localization transition properties in systems
such as monitored free fermions and disordered models.
