  The Uhlmann phase, which reflects the holonomy as the purified state of a
density matrix traverses a loop in the parameter space, has been used to
characterize topological properties of several systems at finite temperatures.
We test the Uhlmann holonomy against various system-environment couplings in
quantum dynamics described by the Lindblad equations of three topological
systems, including the Su-Schrieffer-Heeger (SSH) model, Kitaev chain, and
Bernevig-Hughes-Zhang (BHZ) model. The Uhlmann phase is shown to remain
quantized in all the examples if the initial state is topological and only
certain types of the Lindblad jump operators are present. Topological
protection at finite temperatures against environmental effects in quantum
dynamics is therefore demonstrated albeit only for a restricted class of
system-environment couplings.
