Manuscript En02-E@2 · En02-E-v2.tex · section · c2769986

Declared encoder and dilation

Nonvacuum-calibrated recovery benchmark › Declared encoder and dilation · lines 987–1031

En02-E@2#eq:Nbench

The message is a qubit. On five qubits, define

∣ Φ_{2} ⟩_{ij} = (∣ 00 ⟩_{ij} + ∣ 11 ⟩_{ij}) / 2,

U V ∣ ψ ⟩ F = Y_{2} CX_{3 \to 2} CX_{1 \to 2} (H_{1} S_{1}^{†}) (H_{3} S_{3}^{†}) CZ_{1, 2} CZ_{2, 3}, = (U^{†}_{123} \otimes I_{45}) (∣ ψ ⟩_{2} \otimes ∣ Φ_{2} ⟩_{14} \otimes ∣ Φ_{2} ⟩_{35}), = CZ_{1, 2}, V_{L} = F^{L} V, A_{L} (X) = Tr_{45} (V_{L} X V_{L}^{†}) .

Gate products act rightmost first.

A physical operational packet has basis ${∣ vac ⟩, ∣ 1, 0 ⟩, ∣ 1, 1 ⟩}$ , represented by $∣ 00 ⟩, ∣ 10 ⟩, ∣ 11 ⟩$ . Its atomic raw width is two qubits. The unused binary state $∣ 01 ⟩$ is assigned an invalid-input failure treatment.

At horizon three,

N_{a, L} (X) = (1 - a) tr X ∣ vac^{3} ⟩ ⟨ vac^{3} ∣ + a ι A_{L} (X) ι^{†} .

A coherent dilation retains a branch label in the core, with amplitudes $1 - a, a$ , and extracts encoded factors $1, 2, 3$ on the occupied branch. Tracing the labelled core gives (eq:Nbench). This is a declared operational purification, not the physical purifier of the selected Unruh state.

A surrogate occupied-rail energy $ϵ > 0$ and source energy $E_{in} > 3 ϵ$ give core energies $E_{in} - j ϵ$ after $j$ occupied extractions. This finite conservation law is not identified with the Schwarzschild mass.

Displayed formulas in this passage

unlabeled · line 993
$U V ∣ ψ ⟩ F = Y_{2} CX_{3 \to 2} CX_{1 \to 2} (H_{1} S_{1}^{†}) (H_{3} S_{3}^{†}) CZ_{1, 2} CZ_{2, 3}, = (U^{†}_{123} \otimes I_{45}) (∣ ψ ⟩_{2} \otimes ∣ Φ_{2} ⟩_{14} \otimes ∣ Φ_{2} ⟩_{35}), = CZ_{1, 2}, V_{L} = F^{L} V, A_{L} (X) = Tr_{45} (V_{L} X V_{L}^{†}) .$
En02-E@2#eq:Nbench · line 1013
$N_{a, L} (X) = (1 - a) tr X ∣ vac^{3} ⟩ ⟨ vac^{3} ∣ + a ι A_{L} (X) ι^{†} .$
unlabeled · line 990
$∣ Φ_{2} ⟩_{ij} = (∣ 00 ⟩_{ij} + ∣ 11 ⟩_{ij}) / 2,$