Universality classes for purification in nonunitary quantum processes

TL;DR

This paper explores universal scaling behaviors in quantum purification processes and the singular value structure of large random matrix products, identifying distinct universality classes with implications for quantum information and statistical mechanics.

Contribution

It introduces the concept of universality classes for purification and matrix products, linking them to effective statistical mechanics models and broadening understanding of quantum measurement dynamics.

Findings

Universal scaling forms for purification identified

Distinct universality classes characterized by different scaling functions

Connections established between quantum processes and statistical mechanics models

Abstract

We consider universal aspects of two problems: (i) the slow purification of a large number of qubits by repeated quantum measurements, and (ii) the singular value structure of a product ${m_t m_{t-1}\ldots m_1}$ of many large random matrices. Each kind of process is associated with the decay of natural measures of entropy as a function of time or of the number of matrices in the product. We argue that, for a broad class of models, each process is described by universal scaling forms for purification, and that (i) and (ii) represent distinct ``universality classes'' with distinct scaling functions. Using the replica trick, these universality classes correspond to one-dimensional effective statistical mechanics models for a gas of ``kinks'', representing domain walls between elements of the permutation group. (This is an instructive low-dimensional limit of the effective statistical mechanics models for random circuits and tensor networks.) These results apply to long-time purification in spatially local monitored circuit models on the entangled side of the measurement phase transition.