Manuscript Hawk06-A@1 · Hawk06-A.tex · proposition · c59fe67c

Proposition: Logical observable reconstruction

The operational island criterion · lines 517–546 · proof included or linked

Hawk06-A@1#prop:observables, Hawk06-A@1#eq:Heisenberg

An $e$ -accurate operational island certificate gives a unital CP map

K = D^{*} : B (L) \to B (A)

such that, for every Hermitian $X$ ,

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

At $e = 0$ , a faithful star representation of the logical algebra exists on the correctable accessible support:

X ⟼ T (X \otimes I_{s}) T^{†} .

For $e > 0$ , $K$ need not be multiplicative.

Proof. The left side of (eq:Heisenberg) is $∥ (DN)^{*} (X) - X ∥_{\infty}$ . Trace/operator-norm duality and the channel bound give the result. The exact representation follows from (eq:normalform); its unit is the support projection $T T^{†}$ . An arbitrary CP recovery adjoint is not a star homomorphism.

Displayed formulas in this passage

Hawk06-A@1#eq:Heisenberg · line 525
$V^{†} (K (X) \otimes I_{B}) V - X_{\infty} \leq 2 e ∥ X ∥_{\infty} .$
unlabeled · line 521
$K = D^{*} : B (L) \to B (A)$
unlabeled · line 533
$X ⟼ T (X \otimes I_{s}) T^{†} .$

Proposition: Logical observable reconstruction

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