Lemma: Elementary contraction and effect bounds

A CPTP map contracts trace norm on Hermitian inputs. Consequently trace distance contracts under channels, and channel half-diamond distance contracts under fixed CPTP postprocessing. For states $\rho ,\sigma$ and an effect $0\le F\le I$,

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

The same argument applies with a reference because a CPTP map remains positive after tensoring with its identity. For the effect bound, the trace-zero difference $\rho -\sigma$ has positive and negative parts of equal trace ${\Vert \rho -\sigma \Vert }_1/2$. Testing either part against $F$ proves the assertion.