  A framework for data assimilation combining aspects of operator-theoretic
ergodic theory and quantum mechanics is developed. This framework adapts the
Dirac--von Neumann formalism of quantum dynamics and measurement to perform
sequential data assimilation (filtering) of a partially observed,
measure-preserving dynamical system, using the Koopman operator on the $L^2$
space associated with the invariant measure as an analog of the Heisenberg
evolution operator in quantum mechanics. In addition, the state of the data
assimilation system is represented by a trace-class operator analogous to the
density operator in quantum mechanics, and the assimilated observables by
self-adjoint multiplication operators. An averaging approach is also
introduced, rendering the spectrum of the assimilated observables discrete, and
thus amenable to numerical approximation. We present a data-driven formulation
of the quantum mechanical data assimilation approach, utilizing kernel methods
from machine learning and delay-coordinate maps of dynamical systems to
represent the evolution and measurement operators via matrices in a data-driven
basis. The data-driven formulation is structurally similar to its
infinite-dimensional counterpart, and shown to converge in a limit of large
data under mild assumptions. Applications to periodic oscillators and the
Lorenz 63 system demonstrate that the framework is able to naturally handle
highly non-Gaussian statistics, complex state space geometries, and chaotic
dynamics.
