Condensed Matter/Materials & Biological Physics Seminar with Laimei Nie on Quantifying the Complexity of an Operator

How thermal physics arises in isolated quantum many-body systems has emerged as a central question in condensed matter physics. A key ingredient of quantum thermalization is the growth of operator complexity, which describes the process where a "simple" operator becomes increasingly complicated under the Heisenberg time evolution of a chaotic dynamics. An explicit way to quantify the complexity of an operator is the Shannon entropy of its 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 talk, Nie will examine the validity of coefficient entropy in the presence of hidden symmetries. Using the quantum cat map as an example, he will show that under a generic choice of operator basis, the coefficient entropy fails to capture the suppression of operator growth caused by the symmetries. Nie will 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.