Ensemble phase transitions in measurement-unitary hybrid quantum circuits

arXiv:2611.17322

Saptarshi Mandal, Sthitadhi Roy

The universal properties of state ensembles generated by quantum dynamics provide a refined lens into Hilbert-space ergodicity and deep thermalisation, or lack thereof. We investigate deep-ergodicity breaking transitions of an ancillary qubit entangled with a system undergoing dynamics via hybrid quantum circuits, where unitary evolution is interspersed with projective measurements. Specifically, we analyse the statistical properties of two distinct state ensembles: the trajectory ensemble (TE) conditioned on the measurement outcomes of the hybrid circuit, and the projected ensemble (PE) generated by further projective measurements of the system at the final temporal boundary. We find that both ensembles undergo non-analytic changes -- ensemble phase transitions -- at the critical point of the measurement-induced purification transition, despite their first moments remaining completely maximally mixed and blind to the transition. For Haar-random circuits, the TE transitions from a trivial identity ensemble to the uniform Haar ensemble, while the PE transitions from the Haar ensemble to a trivial singleton pure state. In Clifford circuits, absence of non-stabiliserness limits the PE to a $1$-design, while the TE forms a $k$-design for $k \leq 3$. We demonstrate that these genuine ensemble transitions offer a clear physical mechanism for the measurement-induced entanglement or purification transitions, illustrating how the capacity to ergodically steer a quantum subsystem shifts from the temporal boundary measurements in the mixed phase to the bulk trajectory measurements in the pure phase.