Efficient detection of localization transitions using predictability

TL;DR

This work introduces a complete complementarity framework connecting coherence, predictability, and entanglement for bipartite pure states and demonstrates that predictability, $P_{l1}$, serves as a robust and experimentally efficient marker for localization transitions in both Anderson localization and many-body localization. By analyzing time evolution under disordered, interacting Hamiltonians and employing disorder averaging, the authors show that $P_{l1}$ saturates in Anderson-localized regimes while exhibiting logarithmic changes in MBL, mirroring known coherence and entanglement signatures but with substantially reduced measurement overhead. The study validates both trivial (no bipartition) and bipartite scenarios, and confirms the method’s consistency across several initial states, underscoring predictability as a practical diagnostic for quantum phase transitions. These findings have implications for scalable quantum simulations and offer new avenues to explore quantum information scrambling and ETH-related dynamics with accessible observables.

Abstract

Identifying phase transition points is a fundamental challenge in condensed matter physics, particularly for transitions driven by quantum interference effects, such as Anderson and many-body localization. Recent studies have demonstrated that quantum coherence provides an effective means of detecting localization transitions, offering a practical alternative to full quantum state tomography and related approaches. Building on this idea, we investigate localization transitions through complementarity relations that connect local predictability, local coherence, and entanglement in bipartite pure states. Our results show that predictability serves as a robust and efficient marker for localization transitions. Crucially, its experimental determination requires exponentially fewer measurements than coherence or entanglement, making it a powerful tool for probing quantum phase transitions.

Figures (6)

• Figure 1: An illustration of (a) Anderson and (b) many-body localization, showing single-particle wavefunctions of fermions (blue circles). In (a) there is only a single, non-interacting fermion which is exponentially localized around the site $i = 1$. In (b) we have two fermions in neighboring sites ($i = 1$ and $j = 2$). Even if their single-particle wavefunctions are highly (although not as sharply as in the non-interacting case) localized, leading to lack of transport, the interactions between the particles cause entanglement to spread with time.
• Figure 2: Disorder-averaged time evolution of Eqs. \ref{['CPTriv1']}, \ref{['CPTriv2']} and \ref{['SCR']} for the Hamiltonian \ref{['MBL']} with $J = 1, W = 2$. The values of $g$ (Anderson and MBL cases) are indicated above the subplots.
• Figure 3: Time evolution of the average $2$-site local coherence, predictability, and entanglement entropy under the Hamiltonian \ref{['MBL']} with $J = 1, W = 2$. The values of $g$ are, once again, displayed above the subplots.
• Figure 4: Time evolution of the average $2$-site local coherence, predictability, and entanglement entropy under the Hamiltonian \ref{['MBL']} with $J = 1, W = 2$. The values of $g$ are, once again, displayed above the subplots.
• Figure 5: Disorder-averaged time evolution of Eqs. \ref{['CPTriv1']}, \ref{['CPTriv2']} and \ref{['SCR']} for the Hamiltonian \ref{['MBL']} with $J = 1$. The values of $W$ and $g$ (Anderson and MBL cases) are indicated above the subplots.