  We consider a toy model for the study of monitored dynamics in a many-body
quantum systems. We study the stochastic Schrodinger equation resulting from
the continuous monitoring with a rate $\Gamma$ of a random hermitian operator
chosen at every time from the gaussian unitary ensemble (GUE). Due to
invariance by unitary transformations, the dynamics of the eigenvalues
$\{\lambda_\alpha\}_{\alpha=1}^n$ of the density matrix can be decoupled from
that of the eigenvectors. Thus, stochastic equations are derived that exactly
describe the dynamics of $\lambda$'s. We consider two regimes: in the presence
of an extra dephasing term, which can be generated by imperfect quantum
measurements, the density matrix has a stationary distribution, and we show
that in the limit of large sizes the distribution of $\lambda$'s is described
by an inverse Marchenko Pastur distribution. In the case of perfect
measurements instead, purification eventually occurs and we focus on
finite-time dynamics. In this case, remarkably, we find an exact solution for
the joint probability distribution of $\lambda$'s at each time $t$ and for each
size $n$. Two relevant regimes emerge: at small times $t\Gamma= O(1)$, the
spectrum is in a Coulomb gas regime, with a well-defined continuous spectral
distribution in the limit of $n\to\infty$. In that case, all moments of the
density matrix become self-averaging and it is possible to characterize the
entanglement spectrum exactly. In the limit of large times $t \Gamma = O(n)$
one enters instead a regime in which the eigenvalues are exponentially
separated $\log(\lambda_\alpha/\lambda_\beta) = O(\Gamma t/n)$, but
fluctuations $\sim O(\sqrt{\Gamma t/n})$ play an essential role. We are still
able to characterize the asymptotic behaviors of entanglement entropy in this
regime.
