Fidelity

$F(\rho ,\sigma ):=\mathrm{Tr}[\sqrt{\sqrt{\rho }\sigma \sqrt{\rho }}{]}^2$ . The square roots are well defined as both $\rho$ and $\sigma$ represent quantum states, being positive semidefinite. For pure quantum states $\rho =|\rho ⟩⟨\rho |$ and $\sigma =|\sigma ⟩⟨\sigma |$, we get that $F(\rho ,\sigma )=|⟨\rho |\sigma ⟩{|}^2$.

Properties

• Unitary Group invariance: $F(\rho ,\sigma )=F(U\rho U^{\ast },U\sigma U^{\ast })$ for all Unitary Operators $U$.
• Symmetry: $F(\rho ,\sigma )=F(\sigma ,\rho )$.
• $F(\rho ,\sigma )\in [0,1]$
• $F(\rho ,\sigma )=1\Rightarrow \rho =\sigma$.