Proposition: Joint extraction-domain tail

For every normal referenced input,

For the baseline,

Proof. For one memory, the extraction channel preserves the number block through $a_i$, prepares $a_i$ on $a_i<n\le K_i$, and prepares zero on $n>K_i$. This follows by tracing the distinct memory labels after the swaps.

Both joint channels annihilate cross terms between $P_{1,K_1}\otimes P_{2,K_2}$ and its complement, and agree on the first block. On the complementary positive input block they are CP and have the same trace $p_{\mathbf{K}}$. The half trace norm of the difference of two positive operators of that trace is at most $p_{\mathbf{K}}$. The union bound gives the final assertion.