= Squared quantum fidelity
{title2=$F_{\mathrm{sq}}(\rho,\sigma)=(\operatorname{Tr}\sqrt{\sqrt\rho\sigma\sqrt\rho})^2$}
This is the square of the <unsquared quantum fidelity>, another common convention for <fidelity of quantum states>. If one state is the normalized <pure state> $|\psi\rangle\langle\psi|$, then
$$
F_{\mathrm{sq}}(|\psi\rangle\langle\psi|,\sigma)=\langle\psi|\sigma|\psi\rangle,\qquad
F(|\psi\rangle\langle\psi|,\sigma)=\sqrt{\langle\psi|\sigma|\psi\rangle}.
$$
By the <Born rule>, the <squared quantum fidelity> here is the <probability> that a <quantum measurement> with acceptance <orthogonal projector> $|\psi\rangle\langle\psi|$ accepts $\sigma$. For two <pure states> it is $|\langle\psi|\varphi\rangle|^2$: the comparison of $|0\rangle$ with $|+\rangle$ has acceptance <probability> $1/2$ and <unsquared quantum fidelity> $1/\sqrt2$.
For two general <mixed states>, neither <quantum fidelity> nor <squared quantum fidelity> is a probability that the states are equal. Equality of two specified <density operators> is a definite mathematical condition. The <measurement characterization of quantum fidelity> instead describes overlap between outcome <probability distributions>. The ensemble average used in <reliable quantum source compression> is normally the average <squared quantum fidelity> with the emitted <pure state>; the convention must be stated when comparing formulas.
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