  Recovery of the initial state of a high-dimensional system can require a
large number of measurements. In this paper, we explain how this burden can be
significantly reduced when randomized measurement operators are employed. Our
work builds upon recent results from Compressive Sensing (CS). In particular,
we make the connection to CS analysis for random block diagonal matrices. By
deriving Concentration of Measure (CoM) inequalities, we show that the
observability matrix satisfies the Restricted Isometry Property (RIP) (a
sufficient condition for stable recovery of sparse vectors) under certain
conditions on the state transition matrix. For example, we show that if the
state transition matrix is unitary, and if independent, randomly-populated
measurement matrices are employed, then it is possible to uniquely recover a
sparse high-dimensional initial state when the total number of measurements
scales linearly in the sparsity level (the number of non-zero entries) of the
initial state and logarithmically in the state dimension. We further extend our
RIP analysis for scaled unitary and symmetric state transition matrices. We
support our analysis with a case study of a two-dimensional diffusion process.
