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

TL;DR

This paper studies ensembles of Matrix Product States generated by various quantum circuits, analyzing their spectral properties and revealing a measurement-induced entanglement transition influenced by the circuit dynamics and quantum monitoring.

Contribution

It introduces a framework connecting spectral features of MPS transfer matrices to circuit types and identifies a measurement-induced entanglement transition as a key phenomenon.

Findings

Spectral density features distinguish different circuit types

Ensembles retain physical information about microscopic dynamics

Measurement-induced entanglement transition observed in MPS ensembles

Abstract

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 analyze the spectra of the MPS transfer matrices and relate them to the spreading of mutual information in the MPS state. We demonstrate how different features of the spectral density correspond to distinct types of circuits, revealing that these MPS ensembles retain crucial physical information about the underlying microscopic dynamics. Notably, in the presence of quantum monitoring, we show 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.