  We study joint quasimodes of the Laplacian and one Hecke operator on compact
congruence surfaces, and give conditions on the orders of the quasimodes that
guarantee positive entropy on almost every ergodic component of the
corresponding semiclassical measures. Together with the measure classification
result of the second-named author, this implies Quantum Unique Ergodicity for
such functions. Our result is optimal with respect to the dimension of the
space from which the quasi-mode is constructed.
  We also study equidistribution for sequences of joint quasimodes of the two
partial Laplacians on compact irreducible quotients of
$\mathbb{H}\times\mathbb{H}$.
