Pure-qubit overlap identity ($|⟨u|v⟩{|}^2=\frac{1}{2}(1+u\cdot v)$)

Write each pure density matrix as $(I+r\cdot \sigma )/2$. The trace identities of the Pauli matrices, $\mathrm{Tr}{\sigma }_i=0$ and $\mathrm{Tr}({\sigma }_i{\sigma }_j)=2{\delta }_{ij}$, make the trace of their product equal to the displayed overlap. Antipodal unit vectors represent orthogonal states; vectors at right angles correspond to squared overlap one half. For mixed states the same trace formula gives Hilbert–Schmidt overlap, not in general quantum fidelity.