Manuscript Hawk07-A@1 · Hawk07-A.tex · proposition · f9c0f5da

Proposition: Residual observables and approximate recovery

No duplicate logical exterior · lines 1578–1609 · proof included or linked

Hawk07-A@1#prop:residual

If $C = T_{τ_{B}}$ , then

V^{†} (I_{R} \otimes Z) V = Tr (τ_{B} Z) I_{L} (Z \in B (B)) .

If instead $d_{⋄} (C, T_{τ_{B}}) \leq ζ$ , then

V^{†} (I_{R} \otimes Z) V - Tr (τ_{B} Z) I_{L}_{\infty} \leq 2 ζ ∥ Z ∥_{\infty}

for Hermitian $Z$ .

For the receiver $R$ , an $e$ -accurate decoder has a unital CP adjoint $D^{*}$ satisfying

V^{†} (D^{*} (X) \otimes I_{B}) V - X_{\infty} \leq 2 e ∥ X ∥_{\infty}

for Hermitian logical $X$ . This map need not be a star representation at nonzero error.

Proof. Apply channel trace/operator-norm duality to the complementary comparison and to $DN - id_{L}$ . The exact formula is the zero-error specialization.

Displayed formulas in this passage

unlabeled · line 1581
$V^{†} (I_{R} \otimes Z) V = Tr (τ_{B} Z) I_{L} (Z \in B (B)) .$
unlabeled · line 1587
$V^{†} (I_{R} \otimes Z) V - Tr (τ_{B} Z) I_{L}_{\infty} \leq 2 ζ ∥ Z ∥_{\infty}$
unlabeled · line 1596
$V^{†} (D^{*} (X) \otimes I_{B}) V - X_{\infty} \leq 2 e ∥ X ∥_{\infty}$

Proposition: Residual observables and approximate recovery

Displayed formulas in this passage

Nearby in this section