Proposition: Nested monotonicity

In the release schedule of Section sec:model,

If

then $z_{k+1}(U)\le z_k(U)$.

Proof. Discarding the newly received cells maps ${{\mathcal{N}}_{k+1}}^U$ to ${{\mathcal{N}}_k}^U$. Any earlier decoder can be used later.

The later complement is a partial trace of the earlier complement. Partial trace takes a replacement channel to replacement, so contractivity proves the optimized leakage inequality. It also maps ${{\sigma }_{B_k}}^U$ to ${{\sigma }_{B_{k+1}}}^U$, proving the named-center inequality.