This paper explores symmetry-protected topological (SPT) phases of mixed states in a doubled Hilbert space. The authors study how symmetry and topology interact in quantum many-body mixed states, focusing on short-range entangled (SRE) mixed states of spin systems protected by both average and exact symmetries. By analyzing their pure Choi states in a doubled space, they systematically classify mixed-state SPT (MSPT) phases. The doubled space allows the use of familiar tools for SRE and SPT pure states, though it introduces additional constraints on hermiticity and positivity of the density matrix. The authors investigate the robustness of MSPT invariants under symmetric finite-depth quantum channels, the bulk-boundary correspondence for MSPT phases, and the implications of MSPT invariants for the separability of mixed states and the symmetry-protected sign problem. They also study spontaneous symmetry breaking (SSB) in mixed states, including exact-to-average SSB and order parameters that detect it. The paper discusses the classification of MSPT phases, their relation to previous studies, and the implications of symmetry-protected topological phases in mixed states. The authors conclude that the doubled space perspective provides a systematic framework for understanding mixed-state SPT phases, while also highlighting the challenges and open questions in this area of research.This paper explores symmetry-protected topological (SPT) phases of mixed states in a doubled Hilbert space. The authors study how symmetry and topology interact in quantum many-body mixed states, focusing on short-range entangled (SRE) mixed states of spin systems protected by both average and exact symmetries. By analyzing their pure Choi states in a doubled space, they systematically classify mixed-state SPT (MSPT) phases. The doubled space allows the use of familiar tools for SRE and SPT pure states, though it introduces additional constraints on hermiticity and positivity of the density matrix. The authors investigate the robustness of MSPT invariants under symmetric finite-depth quantum channels, the bulk-boundary correspondence for MSPT phases, and the implications of MSPT invariants for the separability of mixed states and the symmetry-protected sign problem. They also study spontaneous symmetry breaking (SSB) in mixed states, including exact-to-average SSB and order parameters that detect it. The paper discusses the classification of MSPT phases, their relation to previous studies, and the implications of symmetry-protected topological phases in mixed states. The authors conclude that the doubled space perspective provides a systematic framework for understanding mixed-state SPT phases, while also highlighting the challenges and open questions in this area of research.