Manuscript Hawk06-A@1 · Hawk06-A.tex · section · c59fe67c

The complete finite model

The complete finite model · lines 270–337

Hawk06-A@1#eq:Vk, Hawk06-A@1#eq:N, Hawk06-A@1#eq:C

Fix finite spaces

L ≃ C^{d}, H ≃ C^{h}, F ≃ C^{f}, E ≃ C^{h},

where $d \geq 2$ and $h, f \geq 1$ . The diary $L$ is variable. The other input data are fixed.

Prepare

∣ Φ_{h} ⟩_{H E} = \frac{1}{h} a = 1 \sum^{h} ∣ a ⟩_{H} ∣ a ⟩_{E}, ∣ 0 ⟩_{F},

independently of the diary and of every external reference. A fixed embedding unitary $J$ may encode $L H F$ into the vault. A fixed unitary $U$ then acts on the complete vault, of dimension

D = d h f .

For the Haar analysis, $U J$ is one Haar unitary on this $D$ -dimensional space. We absorb $J$ into $U$ .

Choose in advance a tensor-cell presentation of the vault and a nested release schedule. At cut $k$ , write

H_{vault} = R_{k} \otimes B_{k}, r_{k} = dim R_{k}, b_{k} = dim B_{k}, r_{k} b_{k} = D .

Later cuts only release additional cells. No emitted cell or early register is subsequently scrambled by the vault.

The complete prepared isometry is

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

with its factors regrouped into

A_{k} \otimes B_{k}, A_{k} = R_{k} E .

Thus the declared coherent access is $A_{k}$ , not $R_{k}$ alone.

Define

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

These channels preserve the mathematical meaning of every reference extension.

The early register has constant marginal $I_{E} / h$ for every input. At cut zero it does not by itself reveal the diary.

Displayed formulas in this passage

Hawk06-A@1#eq:Vk · line 310
$V_{k}^{U} ∣ ψ ⟩_{L} = (U \otimes I_{E}) (∣ ψ ⟩_{L} ∣ Φ_{h} ⟩_{H E} ∣ 0 ⟩_{F}),$
Hawk06-A@1#eq:N · line 324
$N_{k}^{U} (X) C_{k}^{U} (X) = Tr_{B_{k}} (V_{k}^{U} X (V_{k}^{U})^{†}), = Tr_{A_{k}} (V_{k}^{U} X (V_{k}^{U})^{†}) .$
Hawk06-A@1#eq:C · line 324
$N_{k}^{U} (X) C_{k}^{U} (X) = Tr_{B_{k}} (V_{k}^{U} X (V_{k}^{U})^{†}), = Tr_{A_{k}} (V_{k}^{U} X (V_{k}^{U})^{†}) .$
unlabeled · line 274
$L ≃ C^{d}, H ≃ C^{h}, F ≃ C^{f}, E ≃ C^{h},$
unlabeled · line 284
$∣ Φ_{h} ⟩_{H E} = \frac{1}{h} a = 1 \sum^{h} ∣ a ⟩_{H} ∣ a ⟩_{E}, ∣ 0 ⟩_{F},$
unlabeled · line 294
$D = d h f .$
unlabeled · line 302
$H_{vault} = R_{k} \otimes B_{k}, r_{k} = dim R_{k}, b_{k} = dim B_{k}, r_{k} b_{k} = D .$
unlabeled · line 318
$A_{k} \otimes B_{k}, A_{k} = R_{k} E .$

The complete finite model

Displayed formulas in this passage

Nearby in this section