For questions about the fidelity between quantum states.
Fidelity between two pure states $|{\psi}\rangle$ and $|\phi\rangle$ is defined as: $$\tag{1}F(|{\psi}\rangle, |{\phi}\rangle) = |\langle \psi | \phi \rangle|^2.$$
Fidelity between a pure states $|{\psi}\rangle$ and a mixted state $\rho$ is defined as: $$\tag{2}F(|{\psi}\rangle, \rho) = \langle \psi |\rho| \psi \rangle.$$
Fidelity between two mixed states $\rho$ and $\sigma$ is defined as: $$\tag{3} \begin{align} F(\rho, \sigma) &= || \sqrt{\rho}\sqrt{\sigma} ||_1^2 = \text{Tr}\bigg( \sqrt{\sqrt{\rho}\cdot\sigma\cdot \sqrt{\rho}} \bigg) = \text{Tr}\bigg( \sqrt{\sqrt{\sigma}\cdot\rho\cdot \sqrt{\sigma}} \bigg). \end{align} $$
