Lemma: Variational visibility

For every pair,

Visibility is nondecreasing when $S$ is enlarged.

Proof. For a square matrix $A$, polar decomposition and trace duality give

Apply this identity to ${{\Gamma }^S}_{r{r}^{\prime }}$ and use the defining property of partial trace. The overlap of two unit vectors has modulus at most one. A unitary on a smaller interface extends by the identity to a larger one, proving monotonicity.