Past exam of the mathematics course of the University of Cambridge / 2015 / iii / Paper 66 / 3 / iv / Solution

For two density operators, the trace distance and the unsquared quantum fidelity areThe unsquared convention matters: if $P=|\psi ⟩⟨\psi |$ is a pure state, then $\sqrt{\rho }P\sqrt{\rho }$ is a rank-one positive operator with its sole nonzero eigenvalue equal to $⟨\psi |\rho |\psi ⟩$. Hence