  We introduce a partial state fidelity approach to quantum phase transitions.
We consider a superconducting lattice with a magnetic impurity inserted at its
centre, and look at the fidelity between partial (either one-site or two-site)
quantum states. In the vicinity of the point of the quantum phase transition,
we observe a sudden drop of the fidelity between two one-site partial states
corresponding to the impurity location and its close vicinity. In the case of
two-site states, the fidelity reveals the transition point as long as one of
the two electron sites is located at the impurity, while the other lies
elsewhere in the lattice. We also determine the Uhlmann mixed state geometric
phase, recently introduced in the study of the structural change of the system
state eigenvectors in the vicinity of the lines of thermal phase transitions,
and find it to be trivial, both for one- and two-site partial states, except
when an electron site is at the impurity. This means that the system partial
state eigenvectors do not contribute significantly to the enhanced state
distinguishability around the point of this quantum phase transition. Finally,
we use the fidelity to analyze the total amount of correlations contained
within a composite system, showing that, even for the smallest two-site states,
it features an abrupt quantitative change in the vicinity of the point of the
quantum phase transition.
