Manuscript En07-A@1 · En07-A.tex · theorem · 44229ce9

Theorem: Complete preservation and restoration

Complete channels, access, and local recovery facts › Complete-access restoration is a decoder with a larger input · lines 417–457 · proof included or linked

En07-A@1#thm:global, En07-A@1#eq:restore

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For an isometry $V : C \to R \otimes B$ , the map

L (Y) = V^{†} Y V + Tr ((I - V V^{†}) Y) τ_{C}

is CPTP and satisfies

L V = id_{C} .

For every code–reference trace-class operator $X$ ,

(V \otimes I) X (V^{†} \otimes I)_{1} = ∥ X ∥_{1} .

In particular, if

V ∣ ψ ⟩ = U (J ∣ ψ ⟩ \otimes ∣ Ω ⟩),

where $J$ is an isometry, $∣ Ω ⟩$ is fixed and normalized, and $U$ is a complete unitary, the entire code is preserved.

Proof. The first term of (eq:restore) is completely positive. The second is a positive-effect measurement followed by fixed preparation and is also completely positive. Their traces sum to $Tr Y$ . On the range of $V$ , the second term vanishes and the first returns $X$ .

An isometric embedding preserves all nonzero singular values, including after tensoring with a reference identity. This proves the trace-norm equality. The specified preparation and unitary compose to an isometry.

Displayed formulas in this passage

En07-A@1#eq:restore · line 420
$L (Y) = V^{†} Y V + Tr ((I - V V^{†}) Y) τ_{C}$
unlabeled · line 428
$L V = id_{C} .$
unlabeled · line 432
$(V \otimes I) X (V^{†} \otimes I)_{1} = ∥ X ∥_{1} .$
unlabeled · line 437
$V ∣ ψ ⟩ = U (J ∣ ψ ⟩ \otimes ∣ Ω ⟩),$

Theorem: Complete preservation and restoration

Displayed formulas in this passage

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