Manuscript V@1 · V.tex · theorem · e7ddf37d

Theorem: Mixed coherent witness

The one-oscillator mixed insertion › Same-scalar coherent displacement · lines 2932–2969 · proof included or linked

V@1#thm:coherent

There exist compact real $f, g$ and a nonzero scalar Schwartz vector $z$ such that, for every real $s \neq = 0$ , the preparations

Ψ_{0} = E (0) \otimes E (z), Ψ_{s} = E (s / 2) \otimes E (z)

satisfy

⟨ Ψ_{s}, A_{f}^{g} (κ) Ψ_{s} ⟩ - ⟨ Ψ_{0}, A_{f}^{g} (κ) Ψ_{0} ⟩ = κ s d, d > 0.

The scalar preparation is the same in both states. The choice can have $ϕ (f) = 0$ .

Proof. Apply Theorem thm:externalwitness with $h = e$ . It supplies $f = P ρ$ and compact $g$ for which $v_{e} = K Q_{g e} u_{f} \neq = 0$ . Take $z = v_{e}$ . Then

⟨ Φ (v_{e}) ⟩_{E (z)} = 2 Re ⟨ v_{e}, z ⟩ = 2 ∥ v_{e} ∥^{2} =: d > 0.

For the real oscillator displacement,

⟨ Q ⟩_{E (s / 2)} = s, ⟨ Π ⟩_{E (s / 2)} = 0.

Both means are zero at oscillator vacuum. Use (eq:mixed). All expectations are legitimate by the graph extension in Theorem thm:ESA.

Displayed formulas in this passage

unlabeled · line 2937
$Ψ_{0} = E (0) \otimes E (z), Ψ_{s} = E (s / 2) \otimes E (z)$
unlabeled · line 2942
$⟨ Ψ_{s}, A_{f}^{g} (κ) Ψ_{s} ⟩ - ⟨ Ψ_{0}, A_{f}^{g} (κ) Ψ_{0} ⟩ = κ s d, d > 0.$
unlabeled · line 2956
$⟨ Φ (v_{e}) ⟩_{E (z)} = 2 Re ⟨ v_{e}, z ⟩ = 2 ∥ v_{e} ∥^{2} =: d > 0.$
unlabeled · line 2962
$⟨ Q ⟩_{E (s / 2)} = s, ⟨ Π ⟩_{E (s / 2)} = 0.$

Theorem: Mixed coherent witness

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