Manuscript XXII-F@1 · XXII-F.tex · section · 67821705

Complete-channel approximation corollary

Complete-channel approximation corollary · lines 1132–1184

XXII-F@1#eq:full-target, XXII-F@1#eq:full-error, XXII-F@1#eq:logical-error

Histories, quantum memories, and retained residuals belong in the step interfaces. Hidden correlated memory is not an independent fresh error allowance. The same stability architecture is printed in En08-B and in complete-history comparisons CheungEn08B (lem:telescoping) CheungHawk03A (lem:telescoping).

For a retained output system $F_{n, ζ}$ , define the full target

Θ_{n, ζ} (X) = X \otimes σ_{n, ζ},

where $σ_{n, ζ}$ is a specified input-independent state. It can include an arrival register, history, residual, and failures. The comparison is on this entire declared retained interface, not automatically on every unretained physical environment.

Let

Λ_{n, ζ} = Tr_{F_{n, ζ}} Γ_{n, ζ} .

After logical identification, its target is $id_{M}$ . The unconditional reference marginal is unchanged by every channel, including a replacement channel. This fact alone does not establish identity transport or preservation of message-reference correlations CheungXXIIB,CheungXXIID.

Define

Err_{n}^{full} (ζ) Err_{n}^{log} (ζ) = \frac{∥ a _{n, ζ} - T _{D} ι _{D} ( p ) ∥ _{*} + ∥ x _{n, ζ} - x ∥ _{2} + ∣ τ _{n, ζ} - τ ∣}{ℓ} + d_{⋄} (Γ_{n, ζ}, Θ_{n, ζ}), = \frac{∥ a _{n, ζ} - T _{D} ι _{D} ( p ) ∥ _{*} + ∥ x _{n, ζ} - x ∥ _{2} + ∣ τ _{n, ζ} - τ ∣}{ℓ} + d_{⋄} (Λ_{n, ζ}, id_{M}) .

The source term excludes comparison with an incorrectly anchored protocol.

Displayed formulas in this passage

XXII-F@1#eq:full-error · line 1162
$Err_{n}^{full} (ζ) Err_{n}^{log} (ζ) = \frac{∥ a _{n, ζ} - T _{D} ι _{D} ( p ) ∥ _{*} + ∥ x _{n, ζ} - x ∥ _{2} + ∣ τ _{n, ζ} - τ ∣}{ℓ} + d_{⋄} (Γ_{n, ζ}, Θ_{n, ζ}), = \frac{∥ a _{n, ζ} - T _{D} ι _{D} ( p ) ∥ _{*} + ∥ x _{n, ζ} - x ∥ _{2} + ∣ τ _{n, ζ} - τ ∣}{ℓ} + d_{⋄} (Λ_{n, ζ}, id_{M}) .$
XXII-F@1#eq:logical-error · line 1162
$Err_{n}^{full} (ζ) Err_{n}^{log} (ζ) = \frac{∥ a _{n, ζ} - T _{D} ι _{D} ( p ) ∥ _{*} + ∥ x _{n, ζ} - x ∥ _{2} + ∣ τ _{n, ζ} - τ ∣}{ℓ} + d_{⋄} (Γ_{n, ζ}, Θ_{n, ζ}), = \frac{∥ a _{n, ζ} - T _{D} ι _{D} ( p ) ∥ _{*} + ∥ x _{n, ζ} - x ∥ _{2} + ∣ τ _{n, ζ} - τ ∣}{ℓ} + d_{⋄} (Λ_{n, ζ}, id_{M}) .$
XXII-F@1#eq:full-target · line 1142
$Θ_{n, ζ} (X) = X \otimes σ_{n, ζ},$
unlabeled · line 1152
$Λ_{n, ζ} = Tr_{F_{n, ζ}} Γ_{n, ζ} .$

Complete-channel approximation corollary

Displayed formulas in this passage

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