Theorem: Channel-interface theorem

For a fixed tower, record set $J$, and channel $\Lambda$, the following are equivalent:

1. The coherent and $J$-dephased models agree on every effect available after $\Lambda$.

2. ${\delta }_J(\Lambda )=0$.

3. $\Lambda (X_J)=0$.

Moreover,

The supremum is attained by $P_{+}(\Lambda (X_J))$.

Proof. The operator $\Lambda (X_J)$ is Hermitian and traceless. For every output effect,

Apply Lemma lem:helstrom to $\Lambda (X_J)$. The three conditions are equivalent because the resulting maximum vanishes exactly when that operator vanishes.