= Fidelity of quantum states
{title2=$F(\rho,\sigma)$}
{wiki}
= Quantum fidelity
{synonym}
= Unsquared quantum fidelity
{synonym}
For <density operators> $\rho,\sigma$ on the same finite-dimensional <Hilbert space>, quantum fidelity measures how close their <quantum states> are. This article uses the unsquared convention:
$$
F(\rho,\sigma)=\operatorname{Tr}\sqrt{\sqrt\rho\,\sigma\sqrt\rho}=\|\sqrt\rho\sqrt\sigma\|_1,\qquad 0\leq F(\rho,\sigma)\leq1.
$$
Each square root is the <positive square root of an operator>, and $\|\cdot\|_1$ is the <trace norm>. <Quantum fidelity> is symmetric in its inputs; it is one exactly when $\rho=\sigma$, and zero exactly when $\rho\sigma=0$, meaning that the <support of a positive operator> for one state is <orthogonal> to that for the other. Larger <quantum fidelity> means greater similarity. The separate <squared quantum fidelity> is $F_{\mathrm{sq}}=F^2$. https://jhwatrous.github.io/TQI.3.pdf#page=16[Watrous, The Theory of Quantum Information, Definition 3.11 and Proposition 3.12]{ref}
For two normalized <pure states>, <quantum fidelity> is the magnitude of their <inner product>, $F=|\langle\psi|\varphi\rangle|$. Thus $|0\rangle$ compared with itself has $F=1$, while $|0\rangle$ and $|1\rangle$ have $F=0$. For $|+\rangle=(|0\rangle+|1\rangle)/\sqrt2$, comparison with $|0\rangle$ gives $F=1/\sqrt2$, $F_{\mathrm{sq}}=1/2$, and <trace distance> $D=1/\sqrt2$ by the <pure-state trace distance and fidelity identity>.
Why compare <density operators> rather than their entries? A change of coordinates should not make the same physical preparations look more or less alike. <Unitary invariance of trace distance and fidelity> makes <quantum fidelity> independent of a common basis choice. Multiplying a <pure state> vector by a <global phase> leaves its <density operator> unchanged, so its <quantum fidelity> with the original state is one. These features make <quantum fidelity> useful for checking whether <quantum channels>, state preparations, and <Schumacher compression> preserve a target <quantum state>.
The connection to <Uhlmann's theorem> supplies another beautiful interpretation: <quantum fidelity> is the largest absolute <inner product> between pure vectors obtained from a <purification of a density operator> for each state, using a sufficiently large common reference space. Uhlmann's 1976 paper extended transition probability to algebraic states, including those of a <C-star algebra>; for <density operators> its transition probability corresponds to our <squared quantum fidelity>. The <measurement characterization of quantum fidelity> gives a complementary description directly in terms of observable outcome statistics. https://www.physik.uni-leipzig.de/~uhlmann/PDF/Uh76a.pdf[Uhlmann, The transition probability in the state space of a star-algebra, Reports on Mathematical Physics 9, 273–279, 1976]{ref}
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