Optimal detection of quantum states via projective measurements

We consider the quantum dynamical evolution of a fully-connected quantum system subjected to random projective measurements and study the first detection time of an extended subspace of the Hilbert space. Exact analytical expressions are obtained for the mean first detection time and the full first detection probability distribution as functionals of the initial state $\mathrm{\mid }{\psi }_0⟩$ and the measurement rate $r$. Exact enumeration agrees perfectly with our analytical predictions.