Operator Growth and Symmetry-Resolved Coefficient Entropy in Quantum Maps

Abstract: Operator growth, or operator spreading, describes the process where a "simple" operator acquires increasing complexity under the Heisenberg time evolution of a chaotic dynamics, therefore has been a key concept in the study of quantum chaos in both single-particle and many-body systems. An explicit way to quantify the complexity of an operator is the Shannon entropy of its operator coefficients over a chosen set of operator basis, dubbed "coefficient entropy". However, it remains unclear if the basis-dependency of the coefficient entropy may result in a false diagnosis of operator growth, or the lack thereof. In this paper, we examine the validity of coefficient entropy in the presence of hidden symmetries. Using the quantum cat map as an example, we show that under a generic choice of operator basis, the coefficient entropy fails to capture the suppression of operator growth caused by the symmetries. We further propose "symmetry-resolved coefficient entropy" as the proper diagnosis of operator complexity, which takes into account robust unknown symmetries, and demonstrate its effectiveness in the case of quantum cat map.