Manuscript Hawk04-B@2 · Hawk04-B.tex · proposition · ba3b0713

Proposition: Energy and external readout in the control

The commuting four-Majorana control › An exact frozen realization · lines 2048–2067 · proof included or linked

Hawk04-B@2#prop:control-energy

For this realization,

E_{S} (t) = 0, E_{B} (t) = g^{2} \int_{μ}^{Λ} \frac{1 - cos ( ω t )}{ω ^{2}} d ω, E_{I} (t) = - E_{B} (t) .

Every unobserved residual readout-and-drop leaves (eq:cq) and (eq:Rerror) unchanged.

Proof. The source stays in its zero-energy logical sector. The coherent bath energy is $h_{B}^{1/2} η_{i} (t)^{2}$ , giving the integral. Initial total energy is zero, so conservation gives $E_{I} = - E_{B}$ . The last statement is Theorem thm:drop.

Displayed formulas in this passage

unlabeled · line 2051
$E_{S} (t) = 0, E_{B} (t) = g^{2} \int_{μ}^{Λ} \frac{1 - cos ( ω t )}{ω ^{2}} d ω, E_{I} (t) = - E_{B} (t) .$

Nearby in this section

section: An exact frozen realization
theorem: Theorem: Exact radiation, residual recovery, and complete leakage
section: An exact frozen realization
section: An exact frozen realization