Proof

Proof. The decoded map is

Each factor is a Weyl mixture with identity weight $1-c_d{q}_{\ell }$. The product mixture has identity weight ${\prod }_{\ell }(1-c_d{q}_{\ell })$. Separating that term gives the upper bound (eq:manyexact).

For the lower bound, use a product of $K$ Bell states between code cells and reference cells. Let $F$ be the erased-cell subset. Its probability is

On that branch the erased references are independent and maximally mixed. After any decoder the normalized state has form

Its overlap with the full Bell projector is

The total overlap is at most

The Bell test proves the matching unrestricted-decoder lower bound.

The conservative inequalities follow from the union bound.

For leakage, compare with the product of the single-cell comparison states. The orthogonal memory sector $F$ has weight $p_F$ and contains an embedded copy of ${\mathrm{Tr}}_{F^c}X$. Its distance from the corresponding replacement is at most $p_F(1-d^{-2|F|})$, by Lemma lem:replacement in dimension $d^{|F|}$ and partial-trace contractivity. Summing gives (eq:manyleak). The distance to this particular product comparison is attained by the product Bell input, but alternative joint comparison states are not optimized here.

Finally, each emitted/vacuum flag effect is a scalar multiple of the cell-code identity. The product flag effects therefore have probabilities $p_F$ for every input, including entangled inputs. The branch emits $K-|F|$ excitations, giving the count and energy laws.