Correlations, spectra, and entanglement 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 χ. 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 analyze the spectra of the MPS transfer matrices and relate them to the spreading of mutual information in the MPS state. We show how different characteristics of the spectral density correspond to different types of circuits, revealing that these MPS ensembles retain crucial physical information on the underlying microscopic dynamics within the spectral densities of their transfer matrix. Notably, in the presence of quantum monitoring, we show the existence of a measurement-induced entanglement transition (MIPT) at finite bond dimensions, with the averaged dimension of the null space serving as the effective order parameter.