Manuscript Psy02-A@1 · Psy02-A.tex · proposition · d92ce402

Proposition: Why commutator preservation is the wrong requirement

A separately declared reduction model › Channels are not algebra automorphisms · lines 2605–2638 · proof included or linked

Psy02-A@1#prop:not-automorphism

A dissipative channel adjoint need not preserve products or commutators. For qubit dephasing with parameter $0 \leq λ < 1$ ,

Λ^{*} (X) = λ X, Λ^{*} (Y) = λY, Λ^{*} (Z) = Z,

[Λ^{*} (X), Λ^{*} (Y)] = 2 i λ^{2} Z \neq = Λ^{*} ([X, Y]) .

For oscillator attenuation, the enlarged output annihilator still obeys its canonical commutator because the bath contribution restores it.

Proof. The qubit calculation uses $[X, Y] = 2 i Z$ . For attenuation,

[a_{out}, a_{out}^{†}] = η [a, a^{†}] + (1 - η) [b, b^{†}] = 1 .

Discarding the bath and applying a reduced adjoint is a different operation from retaining this enlarged algebra.

Displayed formulas in this passage

unlabeled · line 2610
$Λ^{*} (X) = λ X, Λ^{*} (Y) = λY, Λ^{*} (Z) = Z,$
unlabeled · line 2616
$[Λ^{*} (X), Λ^{*} (Y)] = 2 i λ^{2} Z \neq = Λ^{*} ([X, Y]) .$
unlabeled · line 2630
$[a_{out}, a_{out}^{†}] = η [a, a^{†}] + (1 - η) [b, b^{†}] = 1 .$