  Meaningful topological invariants for mixed quantum states are challenging to
identify as there is no unique way to define them, and most choices do not
directly relate to physical observables. Here, we propose a simple pragmatic
approach to construct topological invariants of mixed states while preserving a
connection to physical observables, by continuously deforming known topological
invariants for pure (ground) states. Our approach relies on expectation values
of many-body operators, with no reference to single-particle (e.g., Bloch)
wavefunctions. To illustrate it, we examine extensions to mixed states of
$U(1)$ geometric (Berry) phases and their corresponding topological invariant
(winding or Chern number). We discuss measurement schemes, and provide a
detailed construction of invariants for thermal or more general mixed states of
quantum systems with (at least) $U(1)$ charge-conservation symmetry, such as
quantum Hall insulators.
