Manuscript Em06-A@1 · Em06-A-v1.tex · theorem · ee9923c6

Theorem: Preparation-response diagnostic

A nonzero example and one evaluated response › The diagnostic is the response of the named observable · lines 1039–1073 · proof included or linked

Em06-A@1#thm:diagnostic, Em06-A@1#eq:diagnostic

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For the same observable as in Theorem thm:main,

\frac{d}{d α}_{α = 0} ω_{ϵ, α} (O_{f, ϵ}) = 2 Re ⟨ v_{f}, ζ ⟩ = O_{f} [h_{cl}; ϕ_{ζ}] .

The equality holds for every $ϵ > 0$ ; hence the response has the stated classical limit.

For the data of Proposition prop:witness, take $ζ = z_{cl}$ . The response is precisely the strictly negative number in (eq:witness).

Proof. The displacement calculation gives the exact affine formula

ω_{ϵ, α} (O_{f, ϵ}) = 2 Re ⟨ v_{f}, z_{cl} ⟩ + α 2 Re ⟨ v_{f}, ζ ⟩ .

Differentiate this identity. The second equality in (eq:diagnostic) follows from (eq:classicalpairing) and (eq:representation). There is no interchange of an uncontrolled derivative and limit. The final assertion follows from $ϕ_{ζ} = ϕ_{cl}$ and Proposition prop:witness.

Displayed formulas in this passage

Em06-A@1#eq:diagnostic · line 1042
$\frac{d}{d α}_{α = 0} ω_{ϵ, α} (O_{f, ϵ}) = 2 Re ⟨ v_{f}, ζ ⟩ = O_{f} [h_{cl}; ϕ_{ζ}] .$
unlabeled · line 1061
$ω_{ϵ, α} (O_{f, ϵ}) = 2 Re ⟨ v_{f}, z_{cl} ⟩ + α 2 Re ⟨ v_{f}, ζ ⟩ .$

Theorem: Preparation-response diagnostic

Displayed formulas in this passage

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