The paper explores the interplay between symmetry and topology in quantum many-body mixed states, focusing on short-range entangled (SRE) mixed states of spin systems. The authors introduce the concept of symmetry-protected topological (SPT) phases in mixed states, characterized by their pure Choi states in a doubled Hilbert space. They systematically classify these SPT phases, considering both exact and average symmetries, and investigate the robustness of these invariants under symmetric finite-depth quantum channels. The study also examines the bulk-boundary correspondence for mixed-state SPT phases and the implications for separability and the symmetry-protected sign problem. Additionally, the paper discusses spontaneous symmetry breaking (SSB) in mixed states, including exact-to-average SSB, and the order parameters that detect it. The authors provide a detailed framework for understanding mixed-state SPT phases and their properties, extending the familiar concepts from pure states to the more complex context of mixed states.The paper explores the interplay between symmetry and topology in quantum many-body mixed states, focusing on short-range entangled (SRE) mixed states of spin systems. The authors introduce the concept of symmetry-protected topological (SPT) phases in mixed states, characterized by their pure Choi states in a doubled Hilbert space. They systematically classify these SPT phases, considering both exact and average symmetries, and investigate the robustness of these invariants under symmetric finite-depth quantum channels. The study also examines the bulk-boundary correspondence for mixed-state SPT phases and the implications for separability and the symmetry-protected sign problem. Additionally, the paper discusses spontaneous symmetry breaking (SSB) in mixed states, including exact-to-average SSB, and the order parameters that detect it. The authors provide a detailed framework for understanding mixed-state SPT phases and their properties, extending the familiar concepts from pure states to the more complex context of mixed states.