  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.
