Correlations, Spectra and Entaglement Transitions in Ensembles of Matrix Product States

We investigate ensembles of Matrix Product States (MPSs) generated by quantum circuit evolution followed by projection onto MPSs with a fixed bond dimension $\chi$. Specifically, we consider ensembles produced by: (i) random sequential unitary circuits, (ii) random brickwork unitary circuits, and (iii) circuits involving both unitaries and projective measurements. In all cases, we characterize the spectra of the MPS transfer matrix and show that, for the first two cases in the thermodynamic limit, they exhibit a finite universal value of the spectral gap in the limit of large $\chi$, albeit with different spectral densities. We show that a finite gap in this limit does not imply a finite correlation length, as the mutual information between two large subsystems increases with $\chi$ in a manner determined by the entire shape of the spectral density. The latter differs for different types of circuits, indicating that these ensembles of MPS retain relevant physical information about the underlying microscopic dynamics. In particular, in the presence of monitoring, we demonstrate the existence of a measurement-induced entanglement transition (MIPT) in MPS ensembles, with the averaged dimension of the transfer matrix's null space serving as the effective order parameter.