  We investigate the measurement-induced entanglement transition in quantum
circuits built upon Dyson's three circular ensembles (circular unitary,
orthogonal, and symplectic ensembles; CUE, COE and CSE). We utilise the
established model of a one-dimensional circuit evolving under alternating local
random unitary gates and projective measurements performed with tunable rate,
which for gates drawn from the CUE is known to display a transition from
extensive to intensive entanglement scaling as the measurement rate is
increased. By contrasting this case to the COE and CSE, we obtain insights into
the interplay between the local entanglement generation by the gates and the
entanglement reduction by the measurements. For this, we combine exact
analytical random-matrix results for the entanglement generated by the
individual gates in the different ensembles, and numerical results for the
complete quantum circuit. These considerations include an efficient rephrasing
of the statistical entangling power in terms of a characteristic entanglement
matrix capturing the essence of Cartan's KAK decomposition, and a general
result for the eigenvalue statistics of antisymmetric matrices associated with
the CSE.
