# Quantum coherence and the localization transition

## TL;DR

This paper explores how quantum coherence measures can serve as indicators of the localization transition in quantum systems, linking resource theory, majorization, and dynamical properties to distinguish ergodic and localized phases.

## Contribution

It introduces coherence measures as tools to detect localization transitions and connects them with dynamical conductivity, providing a new quantum information perspective on ergodic-MBL phenomena.

## Key findings

- Coherence measures relate to escape probability in localization.
- Majorization-based coherence measures detect transition matrix uniformity.
- Differential coherence corresponds to dynamical conductivity.

## Abstract

A dynamical signature of localization in quantum systems is the absence of transport which is governed by the amount of coherence that configuration space states possess with respect to the Hamiltonian eigenbasis. To make this observation precise, we study the localization transition via quantum coherence measures arising from the resource theory of coherence. We show that the escape probability, which is known to show distinct behavior in the ergodic and localized phases, arises naturally as the average of a coherence measure. Moreover, using the theory of majorization, we argue that broad families of coherence measures can detect the uniformity of the transition matrix (between the Hamiltonian and configuration bases) and hence act as probes to localization. We provide supporting numerical evidence for Anderson and Many-Body Localization (MBL). For infinitesimal perturbations of the Hamiltonian, the differential coherence defines an associated Riemannian metric. We show that the latter is exactly given by the dynamical conductivity, a quantity of experimental relevance which is known to have a distinctively different behavior in the ergodic and in the many-body localized phases. Our results suggest that the quantum information-theoretic notion of coherence and its associated geometrical structures can provide useful insights into the elusive nature of the ergodic-MBL transition.