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

Proposition: Finite-time energy accounting

Finite proxies and access-preserving qubit dictionaries › The energy ledger · lines 1592–1629 · proof included or linked

Hawk04-B@2#prop:energy

Back to manuscript

For the vacuum preparation above, define the source, bath, and interaction expectations by $E_{S} (t), E_{B} (t), E_{I} (t)$ . On every finite interval,

E_{S} (t) + E_{B} (t) + E_{I} (t) = E_{0},

where

E_{0} = Tr ((H_{S} \otimes I_{D}) W ρ W^{†}) \in [0, M_{N}] .

Moreover,

0 \leq E_{B} (t) \leq 2 M_{N} + 4 G_{ω}^{2},

and

- E ˙_{S} = - i ⟨[V_{int}, H_{S}]⟩, E ˙_{B} = - E ˙_{S} - E ˙_{I} .

Proof. The initial bath energy and interaction expectation vanish. Autonomous evolution conserves total energy. Taking expectations in (eq:formlower) gives $E_{B} (t) \leq 2 E_{0} + 4 G_{ω}^{2}$ .

The Dyson graph-norm estimates in Theorem thm:regulator give all fixed occupation moments on finite intervals. Since $h_{B}$ is bounded and the source is finite, the field commutators in the displayed currents are controlled by those moments. Finite-particle approximation therefore justifies differentiation of their expectations. The Heisenberg equation gives the source current, and total energy conservation gives the second identity.

Displayed formulas in this passage

unlabeled · line 1597
$E_{S} (t) + E_{B} (t) + E_{I} (t) = E_{0},$
unlabeled · line 1601
$E_{0} = Tr ((H_{S} \otimes I_{D}) W ρ W^{†}) \in [0, M_{N}] .$
unlabeled · line 1605
$0 \leq E_{B} (t) \leq 2 M_{N} + 4 G_{ω}^{2},$
unlabeled · line 1609
$- E ˙_{S} = - i ⟨[V_{int}, H_{S}]⟩, E ˙_{B} = - E ˙_{S} - E ˙_{I} .$

Nearby in this section

section: The energy ledger