 Solution

Represent the pure states by density matrices ${\rho }_u=(I+u\cdot \sigma )/2$ and ${\rho }_v=(I+v\cdot \sigma )/2$. The Pauli matrices obey $\mathrm{Tr}{\sigma }_i=0$ and $\mathrm{Tr}({\sigma }_i{\sigma }_j)=2{\delta }_{ij}$. ConsequentlyOn the other hand, ${\rho }_u=|u⟩⟨u|$ and ${\rho }_v=|v⟩⟨v|$ make that trace equal to $⟨u|v⟩⟨v|u⟩$. Thus the pure-qubit overlap identity isFor mixed states the trace formula still holds, but its left side is then a Hilbert–Schmidt overlap rather than this squared pure-state overlap.