Proposition: Passive-release monotonicity

Suppose each later transfer acts only on the complete residual and returns an additional quantum factor, leaving earlier accessible factors untouched. Then

Proof. The earlier accessible channel is obtained from the later one by tracing the newly returned factor. A later decoder may therefore discard that factor and use an earlier decoder.

The later complete residual channel is a CPTP postprocessing ${\Lambda }_n{\mathcal{C}}_n$. For every replacement comparison,

Take the infimum.