Manuscript Em06-B@1 · Em06-B-v1.tex · section · 61e38859

The displaced preparation is not bounded-resource

One declared measurement benchmark and preparation cost › The displaced preparation is not bounded-resource · lines 1641–1665

Em06-B@1#eq:number-cost, Em06-B@1#eq:cost-limit

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For finite-number residual states, displacement gives

⟨ N ⟩_{ρ_{ϵ}} = n_{ϵ} + \frac{∥ z ∥ ^{2}}{ϵ} + \frac{2}{ϵ} Re ⟨ a (z) ⟩_{ρ_{ϵ}} .

The annihilation bound implies

∣ ⟨ a (z) ⟩_{ρ_{ϵ}} ∣ \leq ∥ z ∥ n_{ϵ} .

Therefore, under $ϵ n_{ϵ} \to 0$ ,

ϵ ⟨ N ⟩_{ρ_{ϵ}} ⟶ ∥ z ∥^{2} .

For a nonzero classical preparation, the expected unscaled occupation consequently grows as $1/ ϵ$ . No bounded-resource implementation follows from concentration.

Displayed formulas in this passage

Em06-B@1#eq:number-cost · line 1644
$⟨ N ⟩_{ρ_{ϵ}} = n_{ϵ} + \frac{∥ z ∥ ^{2}}{ϵ} + \frac{2}{ϵ} Re ⟨ a (z) ⟩_{ρ_{ϵ}} .$
Em06-B@1#eq:cost-limit · line 1657
$ϵ ⟨ N ⟩_{ρ_{ϵ}} ⟶ ∥ z ∥^{2} .$
unlabeled · line 1652
$∣ ⟨ a (z) ⟩_{ρ_{ϵ}} ∣ \leq ∥ z ∥ n_{ϵ} .$