Manuscript Hawk07-A@1 · Hawk07-A.tex · section · f9c0f5da

Nested release and positive cutoff

The finite model and the certificate › Nested release and positive cutoff · lines 318–366

Hawk07-A@1#eq:V, Hawk07-A@1#eq:channels

Choose a tensor-cell presentation in advance. At cut $k$ , regroup the vault as

P_{k} \otimes B_{k}, r_{k} = dim P_{k}, b_{k} = dim B_{k}, r_{k} b_{k} = D .

The accessible receiver is

R_{k} = P_{k} E, dim R_{k} = r_{k} h .

Previously released cells and $E$ are not subsequently acted on by the vault.

Release means handing over control of named tensor factors. It can also be implemented by SWAPs into reserved blank registers. That implementation adds fixed source placeholders, which must remain in the complete bookkeeping. The primary model needs no such additional placeholders: it changes control over existing laboratory cells. No qubit leaves the laboratory.

The complete prepared isometry is

V_{k}^{U} ∣ ψ ⟩ = (U \otimes I_{E}) (∣ ψ ⟩_{L} ∣ Φ_{h} ⟩_{H E} ∣ 0 ⟩_{F}),

regrouped as $R_{k} B_{k}$ . Set

N_{k}^{U} (X) C_{k}^{U} (X) = Tr_{B_{k}} (V_{k}^{U} X (V_{k}^{U})^{†}), = Tr_{R_{k}} (V_{k}^{U} X (V_{k}^{U})^{†}) .

For qubits,

d = 2^{ℓ}, h = 2^{m}, f = 2^{a}, n = ℓ + m + a, r_{k} = 2^{k}, b_{k} = 2^{n - k} .

A positive-residual cutoff is a predeclared $K < n$ . The physical factor $B_{K}$ remains present.

Displayed formulas in this passage

Hawk07-A@1#eq:V · line 341
$V_{k}^{U} ∣ ψ ⟩ = (U \otimes I_{E}) (∣ ψ ⟩_{L} ∣ Φ_{h} ⟩_{H E} ∣ 0 ⟩_{F}),$
Hawk07-A@1#eq:channels · line 350
$N_{k}^{U} (X) C_{k}^{U} (X) = Tr_{B_{k}} (V_{k}^{U} X (V_{k}^{U})^{†}), = Tr_{R_{k}} (V_{k}^{U} X (V_{k}^{U})^{†}) .$
unlabeled · line 322
$P_{k} \otimes B_{k}, r_{k} = dim P_{k}, b_{k} = dim B_{k}, r_{k} b_{k} = D .$
unlabeled · line 327
$R_{k} = P_{k} E, dim R_{k} = r_{k} h .$
unlabeled · line 359
$d = 2^{ℓ}, h = 2^{m}, f = 2^{a}, n = ℓ + m + a, r_{k} = 2^{k}, b_{k} = 2^{n - k} .$