Uhlmann's theorem

Remarks

Proof idea: fix an orthonormal basis $(e_i)$ of $H$ and an isometric copy of it in $K$, and let $\Omega = \sum_i e_i \otimes e_i$. Every purification of $\rho$ has the form $(\sqrt\rho\, U \otimes I)\Omega$ for some unitary $U$, up to an isometry on $K$, and similarly for $\sigma$. Then $\ip{\psi}{\phi}$ becomes $\tr(\sqrt\rho \sqrt\sigma\, W)$ for a unitary $W$. The largest modulus of such a trace is the trace norm $\norm{\sqrt\rho\sqrt\sigma}_1$, reached through the polar decomposition, and its square is $F(\rho, \sigma)$.

Example: if $\rho = \lvert a \rangle\langle a \rvert$ and $\sigma = \lvert b \rangle\langle b \rvert$ are pure, every purification is $a \otimes k$ or $b \otimes k'$ with unit vectors $k, k'$, and the maximum of $\abs{\ip{a}{b}}^2 \abs{\ip{k}{k'}}^2$ is $\abs{\ip{a}{b}}^2 = F(\rho, \sigma)$.

Going deeper

For a student

Many properties of fidelity follow at once from the theorem. Symmetry $F(\rho, \sigma) = F(\sigma, \rho)$ and $F \le 1$ are immediate. Monotonicity under quantum channels follows because a channel has a Stinespring dilation, which maps purifications to purifications. Joint concavity of $\sqrt F$ also follows. In quantum cryptography and in the theory of decoupling, the theorem is used to build a purification of a nearby state close to a given one.