Manuscript VII-G@1 · VII-G.tex · proposition · 54f561e8

Proposition: Exact expectation balance

Balance laws and admissibility without postselection › Operator tests for energy · lines 1311–1355 · proof included or linked

VII-G@1#prop:energy, VII-G@1#eq:energy-identity, VII-G@1#eq:branch-energy

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Expected energy is conserved for every hybrid input if and only if

h \sum Ψ_{h ∣ g}^{*} (E_{h}) = E_{g} (g \in G) .

The branchwise relations

E_{h} K_{h ∣ g, α} = K_{h ∣ g, α} E_{g}

are sufficient.

If the norm of the defect in (eq:energy-identity) is at most $δ$ for every $g$ , the expected one-step error is at most $δ$ , and the accumulated error after $n$ steps is at most $n δ$ for a common observable and uniform defect.

Proof. Expected conservation is $⟨ E, F ρ ⟩ = ⟨ E, ρ ⟩$ . Testing all states supported in one input block gives the operator identity.

Under (eq:branch-energy),

h, α \sum K_{h ∣ g, α}^{†} E_{h} K_{h ∣ g, α} = h, α \sum K_{h ∣ g, α}^{†} K_{h ∣ g, α} E_{g} = E_{g} .

For a defect $D_{g}$ ,

∣ g \sum Tr (D_{g} ρ_{g})∣ \leq g max ∥ D_{g} ∥_{\infty} g \sum Tr ρ_{g} .

Iteration and the triangle inequality give the accumulated bound.

Displayed formulas in this passage

VII-G@1#eq:energy-identity · line 1314
$h \sum Ψ_{h ∣ g}^{*} (E_{h}) = E_{g} (g \in G) .$
VII-G@1#eq:branch-energy · line 1320
$E_{h} K_{h ∣ g, α} = K_{h ∣ g, α} E_{g}$
unlabeled · line 1341
$h, α \sum K_{h ∣ g, α}^{†} E_{h} K_{h ∣ g, α} = h, α \sum K_{h ∣ g, α}^{†} K_{h ∣ g, α} E_{g} = E_{g} .$
unlabeled · line 1350
$∣ g \sum Tr (D_{g} ρ_{g})∣ \leq g max ∥ D_{g} ∥_{\infty} g \sum Tr ρ_{g} .$

Proposition: Exact expectation balance

Displayed formulas in this passage

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