  We generalize the Fubini-Study method for pure-state complexity to generic
quantum states by taking Bures metric or quantum Fisher information metric on
the space of density matrices as the complexity measure. Due to Uhlmann's
theorem, we show that the mixed-state complexity exactly equals the
purification complexity measured by the Fubini-Study metric for purified states
but without explicitly applying any purification. We also find the purification
complexity is non-increasing under any trace-preserving quantum operations. We
also study the mixed Gaussian states as an example to explicitly illustrate our
conclusions for purification complexity.
