  Information theory, rooted in computer science, and many-body physics, have
traditionally been studied as (almost) independent fields. Only recently has
this paradigm started to shift, with many-body physics being studied and
characterized using tools developed in information theory. In our work, we
introduce a new perspective on this connection, and study phase transitions in
models with randomness, such as localization in disordered systems, or random
quantum circuits with measurements. Utilizing information-based arguments
regarding probability distribution differentiation, we bound critical exponents
in such phase transitions (specifically, those controlling the correlation or
localization lengths). We benchmark our method and rederive the well-known
Harris criterion, bounding critical exponents in the Anderson localization
transition for noninteracting particles, as well as classical disordered spin
systems. We then move on to apply our method to many-body localization. While
in real space our critical exponent bound agrees with recent consensus, we find
that, somewhat surprisingly, numerical results on Fock-space localization for
limited-sized systems do not obey our bounds, indicating that the simulation
results might not hold asymptotically (similarly to what is now believed to
have occurred in the real-space problem). We also apply our approach to random
quantum circuits with random measurements, for which we can derive bounds
transcending recent mappings to percolation problems.
