Manuscript Oai280-B@1 · Oai280-B.tex · section · 6d92ab75

Finite-factor subsystem certification

Finite-factor subsystem certification · lines 2129–2168

Operator-algebra and subsystem recovery are established frameworks OperatorQEC,Harlow. The following result supplies a self-contained finite-factor statistical extension needed to avoid a fixed-spectator overclaim.

Let $C = C^{2}$ , let $S = C^{s}$ be a spectator input, and let

Φ : B (C \otimes S) \to B (C)

be a complete implemented channel. The target is

T_{S} = Tr_{S} .

Its error is

e_{sub} (Φ) = d_{⋄} (Φ, T_{S}),

which compares all referenced inputs, not just a fixed spectator preparation.

Let $Q, R$ be input references for $C, S$ . Define the normalized Choi state

J_{Φ} = (id_{QR} \otimes Φ) (P_{2}^{QC} \otimes P_{s}^{R S}) .

Tensor ordering is identified so that

J_{T_{S}} = P_{2}^{Q C_{out}} \otimes I_{R} / s .

Set

F_{sub} (Φ) = Tr ((P_{2}^{Q C_{out}} \otimes I_{R}) J_{Φ}) .

Displayed formulas in this passage

unlabeled · line 2138
$Φ : B (C \otimes S) \to B (C)$
unlabeled · line 2142
$T_{S} = Tr_{S} .$
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$e_{sub} (Φ) = d_{⋄} (Φ, T_{S}),$
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$J_{Φ} = (id_{QR} \otimes Φ) (P_{2}^{QC} \otimes P_{s}^{R S}) .$
unlabeled · line 2160
$J_{T_{S}} = P_{2}^{Q C_{out}} \otimes I_{R} / s .$
unlabeled · line 2164
$F_{sub} (Φ) = Tr ((P_{2}^{Q C_{out}} \otimes I_{R}) J_{Φ}) .$

Finite-factor subsystem certification

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