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We construct an observable mixed state topological order parameter forsymmetry-protected free fermion matter. It resolves the entire table oftopological insulators and superconductors, relying exclusively on the symmetryclass, but not on unitary symmetries. It provides a robust, quantized signalnot only for pure ground states, but also for mixed states in- or out ofthermal equilibrium. Key ingredient is a unitary probe operator, whose phasecan be related to spectral asymmetry, in turn revealing the topologicalproperties of the underlying state. This is demonstrated analytically in thecontinuum limit, and validated numerically on the lattice. The order parameteris experimentally accessible via either interferometry or full countingstatistics, for example, in cold atom experiments.

Mixed state topological order parameters for symmetry protected fermion matter