Detection of 2D SPT phases under decoherence

We propose a bulk order parameter for extracting symmetry-protected topological (SPT) invariants of quantum many-body mixed states on a two dimensional lattice using partial symmetries. The procedure builds on the partial symmetry order parameter recently developed by some of the authors to study SPT phases of pure states and adapts them to the decohered setting. For a symmetry $G=E\times A$ where $E$ is a strong symmetry and $A$ is a weak symmetry, we show that the partial symmetry order parameter detects SPT invariants jointly protected by $E$ and $A$. We demonstrate this explicitly using a class of mixed states obtained from CZX-type models with ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$ symmetry and subjecting them to noise that weakens one of the ${\mathbb{Z}}_2$ symmetries. We also comment on the practical detection of SPT invariants in quantum simulators through randomized measurements.