Theorem: Energy-constrained finite-code route

Let $H_{\mathrm{E}}\ge 0$, let $P_K$ be finite rank, and suppose $H_{\mathrm{E}}\ge {\Delta }_K(1-P_K)$, ${\Delta }_K>0$. Reset the high block to a fixed low-energy vector:

For referenced inputs of energy at most $E$,

If two channels have finite-code error $e_K$, then

The bound may be truncated at one.

Proof. The energy inequality proves the tail estimate. For a purification $\Psi$, projection gives a pure-state versus subnormalized-pure-state trace norm

The reset block contributes trace $t_K$. Partial trace proves the state bound. Insert the reset input in both channel outputs and apply the finite-code bound to the reset state.