Fidelity

Formula

This card uses squared Uhlmann fidelity:

F(\rho,\sigma) =\left[ \operatorname{Tr}\sqrt{\sqrt\rho\,\sigma\sqrt\rho} \right]^2.

Useful special cases are

F(|\psi\rangle\langle\psi|,\sigma) =\langle\psi|\sigma|\psi\rangle,

F(|\psi\rangle,|\phi\rangle)=|\langle\psi|\phi\rangle|^2,

and, for commuting eigenvalue distributions $p_i,q_i$ ,

F(\rho,\sigma)=\left(\sum_i\sqrt{p_iq_i}\right)^2.

For qubit Bloch vectors $\mathbf r,\mathbf s$ ,

F(\rho,\sigma) =\frac12\left[ 1+\mathbf r\cdot\mathbf s +\sqrt{(1-\lVert\mathbf r\rVert^2) (1-\lVert\mathbf s\rVert^2)} \right].

Assumptions and Conventions

$\rho$ and $\sigma$ are normalized density operators on the same space.
Many sources call the square root $\sqrt F$ “fidelity.” Always state which convention is used.
Matrix square roots are the positive square roots from spectral calculus.

Symbols

Symbol	Meaning
$F(\rho,\sigma)$	squared fidelity, between $0$ and $1$
$\sqrt\rho$	positive square root of $\rho$
$\mathbf r,\mathbf s$	qubit Bloch vectors
$D(\rho,\sigma)$	trace distance

Validity and Warnings

$F=1$ exactly when $\rho=\sigma$ ; orthogonal supports give $F=0$ .
Fidelity is not itself a metric.
It is multiplicative on tensor products and monotone upward under a common channel: processing makes states no less similar.
In this convention the Fuchs–van de Graaf inequalities are

1-\sqrt F\le D\le\sqrt{1-F}.

Do not silently clip substantial negative eigenvalues of an estimated state before reporting fidelity.

Canonical Treatment

Purification meaning, operational comparison, commuting and qubit examples, channel behavior, numerical checks, exercises, and references are at Density Operators for Quantum Information.