  Probing the out-of-equilibrium dynamics of quantum matter has gained renewed
interest owing to immense experimental progress in artifcial quantum systems.
Dynamical quantum measures such as the growth of entanglement entropy (EE) and
out-of-time ordered correlators (OTOCs) have been shown, theoretically, to
provide great insight by exposing subtle quantum features invisible to
traditional measures such as mass transport. However, measuring them in
experiments requires either identical copies of the system, an ancilla qubit
coupled to the whole system, or many measurements on a single copy, thereby
making scalability extremely complex and hence, severely limiting their
potential. Here, we introduce an alternate quantity $-$ the out-of-time-ordered
measurement (OTOM) $-$ which involves measuring a single observable on a single
copy of the system, while retaining the distinctive features of the OTOCs. We
show, theoretically, that OTOMs are closely related to OTOCs in a doubled
system with the same quantum statistical properties as the original system.
Using exact diagonalization, we numerically simulate classical mass transport,
as well as quantum dynamics through computations of the OTOC, the OTOM, and the
EE in quantum spin chain models in various interesting regimes (including
chaotic and many-body localized systems). Our results demonstrate that an OTOM
can successfully reveal subtle aspects of quantum dynamics hidden to classical
measures, and crucially, provide experimental access to them.
