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

Concentration does not imply a fluctuation limit

Nonzero success and explicit failure examples › Concentration does not imply a fluctuation limit · lines 1545–1586

Let the residual state alternate between the vacuum and the one-particle state in the mode $e$ , according to the parity of $⌊ lo g (1/ ϵ)⌋$ . Then $n_{ϵ} \leq 1$ , so the relational coefficient concentrates on $o$ .

However, along the two parity subsequences the normalized fluctuation characteristic functions are respectively

e^{- t^{2} ∥ v ∥^{2} /2}, e^{- t^{2} ∥ v ∥^{2} /2} (1 - t^{2} ∥ v ∥^{2}) .

They differ, for example at $t = 1/∥ v ∥$ . There is no unique marginal fluctuation limit.

For a separate moment caution, let

ρ_{ϵ} = (1 - ϵ) ∣ Ω ⟩ ⟨ Ω ∣ + ϵ ∣ n_{ϵ}^{mode} ⟩ ⟨ n_{ϵ}^{mode} ∣, n_{ϵ}^{mode} = ⌊ ϵ^{- 2} ⌋

in the same mode $e$ . This state tends to the vacuum in trace norm, but

⟨ Φ (v)^{2} ⟩_{ρ_{ϵ}} = ∥ v ∥^{2} (1 + 2 ϵ ⌊ ϵ^{- 2} ⌋) ⟶ \infty.

Its normalized fluctuation laws converge weakly to the vacuum law, while their second moments do not. Moreover,

ϵ ⟨ Φ (v)^{2} ⟩_{ρ_{ϵ}} ⟶ 2∥ v ∥^{2},

so Wasserstein concentration on $o$ fails despite the weak classical concentration caused by the vanishing mixture weight.

Displayed formulas in this passage

unlabeled · line 1554
$e^{- t^{2} ∥ v ∥^{2} /2}, e^{- t^{2} ∥ v ∥^{2} /2} (1 - t^{2} ∥ v ∥^{2}) .$
unlabeled · line 1563
$ρ_{ϵ} = (1 - ϵ) ∣ Ω ⟩ ⟨ Ω ∣ + ϵ ∣ n_{ϵ}^{mode} ⟩ ⟨ n_{ϵ}^{mode} ∣, n_{ϵ}^{mode} = ⌊ ϵ^{- 2} ⌋$
unlabeled · line 1573
$⟨ Φ (v)^{2} ⟩_{ρ_{ϵ}} = ∥ v ∥^{2} (1 + 2 ϵ ⌊ ϵ^{- 2} ⌋) ⟶ \infty.$
unlabeled · line 1580
$ϵ ⟨ Φ (v)^{2} ⟩_{ρ_{ϵ}} ⟶ 2∥ v ∥^{2},$