Information and Physics: How information theory could be used to bound physical critical exponents

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. I will start by reviewing several such directions, ranging from Maxwell’s demon and information engines, through information bounds on renormalization group flows, to the black hole information paradox, then to bounds on classical and quantum glass annealing dynamics based on their computational complexity. I will then move forward to our own recent work, and bring a new perspective on this connection, studying phase transitions in models with randomness, such as localization in disordered systems, or random quantum circuits with measurements. Utilizing information-based arguments on probability distribution differentiation, we bound the critical exponents in such phase transitions. We benchmark our method and rederive the well-known Harris criterion, bounding the critical exponent in the Anderson localization transition for noninteracting particles, as well as in classical disordered spin systems, and then extend it to non-Anderson transitions. Moving forward to many-body localization, we infer bounds both on real space and Fock space localization critical exponents. Interestingly, our bounds are not obeyed by prior studies in several cases; some of our bounds are aligned with recent consensus, while some point to newly found problems in previously obtained numerical and analytical results. Additionally, we apply our method to measurement-induced phase transitions in random quantum circuits. To date, analytical results for such phase transitions have only been obtained for one-dimensional circuits with Haar-random unitary gates, using the zeroth Rènyi entropy as a marker of the phases. Our bounds are valid for arbitrary circuit dimensions, gate combinations, and phase markers, and are obeyed by existing numerical results. I will finish by discussing how our approach could be generalized to clean systems and give rise to new bounds on dynamical critical exponents, for either physical or computational (Monte Carlo) classical as well as quantum dynamics.