Manuscript Oai265-A@2 · Oai265-A.tex · lemma · c0a34a45

Lemma: Restoration from the complete isometric output

Stage 5: complete channels and the entropy separation › Complete access and exact accessible recovery · lines 1271–1293 · proof included or linked

Oai265-A@2#lem:global

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Let $τ_{C} = I_{C} / d$ . Then

L (Y) = V^{†} Y V + Tr ((I_{R B} - V V^{†}) Y) τ_{C}

is CPTP and $L Ad_{V} = id_{C}$ . Also, for every reference operator $X$ ,

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

Proof. Compression and effect-measurement followed by fixed-state preparation are CP. Their traces sum to $Tr Y$ . On the encoded range the second term vanishes and $V^{†} V = I$ . Isometric embeddings preserve nonzero singular values, proving the norm identity.

Displayed formulas in this passage

unlabeled · line 1274
$L (Y) = V^{†} Y V + Tr ((I_{R B} - V V^{†}) Y) τ_{C}$
unlabeled · line 1282
$∥ (I_{Q} \otimes V) X (I_{Q} \otimes V^{†}) ∥_{1} = ∥ X ∥_{1} .$

Lemma: Restoration from the complete isometric output

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