  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.
