Measuring operator size growth in quantum quench experiments

Abstract: Operator scrambling denotes the evolution of a simple operator into a complicated one (in the Heisenberg picture), which characterizes quantum chaos in many-body systems. More specifically, a simple operator evolves into a linear superposition of many operators, most of which are many-body operators supported on a region of size much larger than $1$. In general, an operator does not have a definite size but is characterized by a probability distribution of size. The operator size is related to out-of-time-order correlation functions, but these are generically difficult to obtain from experimental observables. In this paper we show that the operator size distribution can be measured in quantum quench experiments. In a quantum spin system, we propose to prepare an ensemble of initial states which are direct product states of random pure states of each spin qudit, and measure a simple physical observable (such as a particular component of spin) at later time $t$. The initial state dependence of the expectation value measures a particular component of the operator size distribution. Furthermore, many other features of the operator size distribution can be measured by analyzing the same data, such as the support of the operator in space.