Lemma: Contraction, effects, and pure-state distance

A CPTP map contracts trace norm on Hermitian inputs and therefore contracts trace distance. Half-diamond distance contracts under fixed CPTP pre- and postprocessing. For states and an effect $0\le F\le I$,

For normalized vectors $x,y$,

In particular,

Proof. For Hermitian $X=X_{+}-X_{-}$, positivity and trace preservation give

The same argument applies with a reference. Preprocessing restricts the referenced input states available in the supremum; postprocessing contracts their output distances.

The positive and negative parts of $\rho -\sigma$ have equal trace ${\Vert \rho -\sigma \Vert }_1/2$, which proves the effect inequality. The difference of two rank-one states has support in their two-dimensional span; its nonzero eigenvalues are $\pm \sqrt{1-|⟨x,y⟩{|}^2}$. Finally, test the Bell-input output against $|\Phi ⟩⟨\Phi |$.