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The mixedness of a quantum state is usually seen as an adversary totopological quantization of observables. For example, exact quantization of thecharge transported in a so-called Thouless adiabatic pump is lifted at anyfinite temperature in symmetry-protected topological insulators. Here, we showthat certain directly observable many-body correlators preserve the integrityof topological invariants for mixed Gaussian quantum states in one dimension.Our approach relies on the expectation value of the many-bodymomentum-translation operator, and leads to a physical observable --- the"ensemble geometric phase" (EGP) --- which represents a bona fide geometricphase for mixed quantum states, in the thermodynamic limit. In cyclicprotocols, the EGP provides a topologically quantized observable which detectsencircled spectral singularities ("purity-gap" closing points) of densitymatrices. While we identify the many-body nature of the EGP as a keyingredient, we propose a conceptually simple, interferometric setup to directlymeasure the latter in experiments with mesoscopic ensembles of ultracold atoms.

Probing the topology of density matrices