Tensor network formulation of symmetry protected topological phases in mixed states

TL;DR

This paper develops a tensor network-based framework to classify symmetry-protected topological (SPT) phases in mixed quantum states, extending the classification from pure states to mixed states in one and two dimensions.

Contribution

It introduces a classification scheme for SPT phases in mixed states using tensor networks, including a complete cohomology-based classification in 1D and extension to 2D.

Findings

Classifies 1D SPT phases in mixed states via cohomology groups.

Extends the classification to 2D mixed states with tensor networks.

Shows the classification is preserved under symmetric local circuits.

Abstract

We define and classify symmetry-protected topological (SPT) phases in mixed states based on the tensor network formulation of the density matrix. In one dimension, we introduce strong injective matrix product density operators (MPDO), which describe a broad class of short-range correlated mixed states, including the locally decohered SPT states. We map strong injective MPDO to a pure state in the doubled Hilbert space and define the SPT phases according to the cohomology class of the symmetry group in the doubled state. Although the doubled state exhibits an enlarged symmetry, the possible SPT phases are also constrained by the Hermiticity and the semi-positivity of the density matrix. We here obtain a complete classification of SPT phases with a direct product of strong $G$ and weak $K$ unitary symmetry given by the cohomology group $H^2(G, \text{U}(1))\oplus H^1(K, H^1(G, \text{U}(1)))$. The SPT phases in our definition are preserved under symmetric local circuits consisting of non-degenerate channels. This motivates an alternative definition of SPT phases according to the equivalence class of mixed states under a ``one-way" connection using symmetric non-degenerate channels. In locally purifiable MPDO with strong symmetry, we prove that this alternative definition reproduces the cohomology classification. We further extend our results to two-dimensional mixed states described by strong semi-injective tensor network density operators and classify the possible SPT phases.