  We develop an algebraic framework for sequential data assimilation of
partially observed dynamical systems. In this framework, Bayesian data
assimilation is embedded in a non-abelian operator algebra, which provides a
representation of observables by multiplication operators and probability
densities by density operators (quantum states). In the algebraic approach, the
forecast step of data assimilation is represented by a quantum operation
induced by the Koopman operator of the dynamical system. Moreover, the analysis
step is described by a quantum effect, which generalizes the Bayesian
observational update rule. Projecting this formulation to finite-dimensional
matrix algebras leads to new computational data assimilation schemes that are
(i) automatically positivity-preserving; and (ii) amenable to consistent
data-driven approximation using kernel methods for machine learning. Moreover,
these methods are natural candidates for implementation on quantum computers.
Applications to data assimilation of the Lorenz 96 multiscale system and the El
Nino Southern Oscillation in a climate model show promising results in terms of
forecast skill and uncertainty quantification.
