Manuscript Hawk02-A@1 · Hawk02-A.tex · proposition · 7fbae745

Proposition: Mean-field feedback can be nonaffine

Stage 2: the backreaction interface › State-dependent geometry does not automatically define a channel · lines 613–638 · proof included or linked

Hawk02-A@1#prop:nonaffine

A rule of the form

ρ ⟼ U_{g [ρ]} ρ U_{g [ρ]}^{†}

is not automatically a quantum channel, even when each $U_{g [ρ]}$ is unitary.

Proof. For a qubit, let $X, Y, Z$ be the Pauli matrices and set

F_{t} (ρ) = e^{- i t Tr (Z ρ) X} ρ e^{i t Tr (Z ρ) X} .

Write $ρ_{0} = ∣ 0 ⟩ ⟨ 0 ∣$ and $ρ_{1} = ∣ 1 ⟩ ⟨ 1 ∣$ . Then $F_{t} (I /2) = I /2$ , but

\frac{F _{t} ( ρ _{0} ) + F _{t} ( ρ _{1} )}{2} = \frac{I - sin ( 2 t ) Y}{2} .

For $sin (2 t) \neq = 0$ , the rule is not affine and therefore not CPTP.

Displayed formulas in this passage

unlabeled · line 616
$ρ ⟼ U_{g [ρ]} ρ U_{g [ρ]}^{†}$
unlabeled · line 625
$F_{t} (ρ) = e^{- i t Tr (Z ρ) X} ρ e^{i t Tr (Z ρ) X} .$
unlabeled · line 632
$\frac{F _{t} ( ρ _{0} ) + F _{t} ( ρ _{1} )}{2} = \frac{I - sin ( 2 t ) Y}{2} .$