Theorem: Three retained cuts with one fixed receiver

Take

Declare

The three cuts have

If

then a set of Haar unitaries of measure at least $1/64$ satisfies all of the following:

1. $z_k(U)\le \zeta$ for every $k_{\ast }\le k\le K$;

2. the fixed decoder policy (eq:fixedpolicy) has half-diamond error at most $\min \{a_{16},\sqrt{2\zeta }\}$ at all three cuts;

3. for the fixed pure diary input,

   $$S({P_{k_{\mathrm{peak}}}}^U)>{\log }_2(2h)\ge \max \{S({P_{k_{\ast }}}^U),S({P_K}^U)\};$$

4. every append-only destructive record is nondecreasing;

5. the cutoff retains one physical vault qubit.

The ensemble mean also rises and falls strictly at these three cuts.

Proof. At $k_{\ast }$, (eq:K) gives

Hence, under (eq:size),

At the intermediate cut,

Markov's inequality gives

Therefore an event of probability greater than $1/32$ has

At $k_{\ast }$, rank is at most $2h$. At $K$, rank is at most $b_{K}h=2h$. Their entropy upper bounds hold for every unitary.

Subtracting the leakage failure set from the turnover event leaves measure greater than $1/64$, hence at least $1/64$. Nested monotonicity and Theorem thm:decoder prove (i)–(ii). Proposition prop:record gives (iv), and $b_K=2$ gives (v).

Finally,

The mean entropy lower bound in Theorem thm:entropy is therefore strictly greater than ${\log }_2(2h)$ at the intermediate cut, whereas both neighboring mean entropies are at most that value.