Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 323 / 4 / a / iv / Solution

The difference $\tau ={\rho }_1-{\rho }_2$ determinesso it suffices for trace distance.

It does not determine fidelity. For $0<t<1$, both pairsandhave the same difference $\mathrm{diag}(t,-t)$. Their fidelities are respectively $\sqrt{1-t^2}$ and $\sqrt{1-t}$, which are generally unequal.