Exact renormalization group flow for matrix product density operators

TL;DR

This paper develops an exact renormalization group framework for a specific subclass of matrix product density operators, revealing their algebraic structure and implications for quantum phase classification.

Contribution

It introduces a subclass of MPDOs with a well-defined exact RG flow and uncovers their pre-bialgebra and symmetry structures, advancing understanding of mixed-state quantum phases.

Findings

Identified conditions for MPDOs to admit an exact RG flow.

Revealed the algebraic structure of MPDOs in the subclass as pre-bialgebras.

Discussed implications for classifying mixed-state quantum phases.

Abstract

Matrix product density operator (MPDO) provides an efficient tensor network representation of mixed states on one-dimensional quantum many-body systems. We study a real-space renormalization group transformation of MPDOs represented by a circuit of local quantum channels. We require that the renormalization group flow is exact, in the sense that it exactly preserves the correlation between the coarse-grained sites and is therefore invertible by another circuit of local quantum channels. Unlike matrix product states (MPS), which always have a well-defined isometric renormalization transformation, we show that general MPDOs do not necessarily admit a converging exact renormalization group flow. We then introduce a subclass of MPDOs with a well-defined renormalization group flow, and show the structure of the MPDOs in the subclass as a representation of a pre-bialgebra structure. As a result, such MPDOs obey generalized symmetry represented by matrix product operator algebras associated with the pre-bialgebra. We also discuss implications with the classification of mixed-state quantum phases.