Random unitary maps for quantum state reconstruction

We study the possibility of performing quantum state reconstruction from a measurement record that is obtained as a sequence of expectation values of a Hermitian operator evolving under repeated application of a single random unitary map, ${U}_{0}$. We show that while this single-parameter orbit in operator space is not informationally complete, it can be used to yield surprisingly high-fidelity reconstruction. For a $d$-dimensional Hilbert space with the initial observable in $\mathfrak{su}(d)$, the measurement record lacks information about a matrix subspace of dimension $\ensuremath{\geqslant}d\ensuremath{-}2$ out of the total dimension ${d}^{2}\ensuremath{-}1$. We determine the conditions on ${U}_{0}$ such that the bound is saturated, and show they are achieved by almost all pseudorandom unitary matrices. When we further impose the constraint that the physical density matrix must be positive, we obtain even higher fidelity than that predicted from the missing subspace. With prior knowledge that the state is pure, the reconstruction will be perfect (in the limit of vanishing noise) and for arbitrary mixed states, the fidelity is over $0.96$, even for small $d$, and reaching $\mathcal{F}g0.99$ for $dg9$. We also study the implementation of this protocol based on the relationship between random matrices and quantum chaos. We show that the Floquet operator of the quantum kicked top provides a means of generating the required type of measurement record, with implications on the relationship between quantum chaos and information gain.