Manuscript XXII-F@1 · XXII-F.tex · section · 67821705

Destructive records are not quantum returns

Readout, limiting tests, and dynamics separation › Destructive records are not quantum returns · lines 1512–1539

XXII-F@1#eq:readout

For a measured factor $S$ , an untouched residual $B$ , and a declared orthonormal readout basis, the deterministic record channel is

M_{S} (X) = z \sum ∣ z ⟩ ⟨ z ∣_{R} \otimes (⟨ z ∣_{S} \otimes I_{B}) X (∣ z ⟩_{S} \otimes I_{B}) .

Its Kraus operators are $∣ z ⟩_{R} \otimes ⟨ z ∣_{S} \otimes I_{B}$ ; their adjoint products sum to $I_{S B}$ . Thus it is CPTP, including reference extensions, and

Tr_{R} M_{S} (X) = Tr_{S} X .

Its outputs are $R, B$ , not the consumed quantum factor $S$ . This is the supplied finite readout convention and consumption discipline of I, En09-A, and XVIII-A CheungI,CheungEn09A,CheungXVIIIA.

The surviving complement can remain quantum and correlated with a reference. It is not a quantum return of the measured factor. A decoder returning the unknown message is a different map with a different correctness requirement. No normalized postselected branch replaces the deterministic channel in the corollary.

Displayed formulas in this passage

XXII-F@1#eq:readout · line 1516
$M_{S} (X) = z \sum ∣ z ⟩ ⟨ z ∣_{R} \otimes (⟨ z ∣_{S} \otimes I_{B}) X (∣ z ⟩_{S} \otimes I_{B}) .$
unlabeled · line 1526
$Tr_{R} M_{S} (X) = Tr_{S} X .$