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

Centered amplified fluctuations prevent concentration

Nonzero success and explicit failure examples › Centered amplified fluctuations prevent concentration · lines 1487–1528

Em06-B@1#eq:squeezed, Em06-B@1#eq:squeezed-scaling

The one-mode Schrödinger representation gives

Φ (e) = 2 x, Φ (i e) = - i 2 \partial_{x}, N_{e} = \frac{1}{2} (x^{2} - \partial_{x}^{2} - 1) .

For real $r$ , define the normalized vector

g_{r} (x) = π^{- 1/4} e^{- r /2} exp (- \frac{1}{2} e^{- 2 r} x^{2}),

with vacuum on $e^{⊥}$ . Elementary Gaussian integration gives

⟨ Φ (e)⟩ = 0, Var (Φ (e)) = e^{2 r}, ⟨ N_{e} ⟩ = \frac{e ^{2 r} + e ^{- 2 r} - 2}{4} = sinh^{2} r .

This is an explicitly constructed Fock-normal squeezed preparation.

Take $r_{ϵ} = \frac{α}{2} lo g (1/ ϵ)$ , $α \geq 0$ . For the observed direction $v$ ,

⟨ O_{ϵ} ⟩ = o, Var (O_{ϵ}) = ϵ^{1 - α} ∥ v ∥^{2} .

Thus:

0 \leq α < 1 α = 1 α > 1 concentration holds, variance is the nonzero constant ∥ v ∥^{2}, variance diverges .

The failure at $α = 1$ is an exact centered, nonvanishing-variance failure of the acceptance pair.

Displayed formulas in this passage

Em06-B@1#eq:squeezed · line 1496
$g_{r} (x) = π^{- 1/4} e^{- r /2} exp (- \frac{1}{2} e^{- 2 r} x^{2}),$
Em06-B@1#eq:squeezed-scaling · line 1512
$⟨ O_{ϵ} ⟩ = o, Var (O_{ϵ}) = ϵ^{1 - α} ∥ v ∥^{2} .$
unlabeled · line 1490
$Φ (e) = 2 x, Φ (i e) = - i 2 \partial_{x}, N_{e} = \frac{1}{2} (x^{2} - \partial_{x}^{2} - 1) .$
unlabeled · line 1502
$⟨ Φ (e)⟩ = 0, Var (Φ (e)) = e^{2 r}, ⟨ N_{e} ⟩ = \frac{e ^{2 r} + e ^{- 2 r} - 2}{4} = sinh^{2} r .$
unlabeled · line 1519
$0 \leq α < 1 α = 1 α > 1 concentration holds, variance is the nonzero constant ∥ v ∥^{2}, variance diverges .$