Manuscript Em03-D@1 · Em03-D-v1.tex · section · 450d878a

An exact finite-plane squeeze

Probe preparation, noise, and squeezing cost › An exact finite-plane squeeze · lines 1389–1435

Em03-D@1#eq:squeeze

The free probe test quotient has symplectic form

s_{ψ} (u, w) = \int u E w d vol_{g} .

Choose a real compact s-wave test $p_{0}$ for which

c_{0} = s_{ψ} (g, p_{0}) \neq = 0.

Such a test exists by nondegeneracy of the free scalar symplectic space. It can also be constructed from compact Cauchy data symplectically paired with the nonzero solution $E g$ , followed by the time-slice construction.

For sufficiently small $ϵ$ , define

p_{ϵ} = \frac{p _{0}}{s _{ψ} ( g _{ϵ} , p _{0} )} .

Then $s_{ψ} (g_{ϵ}, p_{ϵ}) = 1$ .

Let $Π_{ϵ}$ be the symplectic projection onto their real plane:

Π_{ϵ} u = s_{ψ} (u, p_{ϵ}) g_{ϵ} - s_{ψ} (u, g_{ϵ}) p_{ϵ} .

Define a symplectic map on the probe test quotient by

S_{ϵ} u = u - Π_{ϵ} u + ϵ^{2} s_{ψ} (u, p_{ϵ}) g_{ϵ} - ϵ^{- 2} s_{ψ} (u, g_{ϵ}) p_{ϵ} .

It scales $g_{ϵ}$ by $ϵ^{2}$ , scales $p_{ϵ}$ by $ϵ^{- 2}$ , and fixes the symplectic orthogonal complement.

Define the independently prepared probe state by

σ_{R, ϵ} (W_{ψ} (u)) = σ_{ref} (W_{ψ} (S_{ϵ} u)) .

Displayed formulas in this passage

Em03-D@1#eq:squeeze · line 1419
$S_{ϵ} u = u - Π_{ϵ} u + ϵ^{2} s_{ψ} (u, p_{ϵ}) g_{ϵ} - ϵ^{- 2} s_{ψ} (u, g_{ϵ}) p_{ϵ} .$
unlabeled · line 1392
$s_{ψ} (u, w) = \int u E w d vol_{g} .$
unlabeled · line 1396
$c_{0} = s_{ψ} (g, p_{0}) \neq = 0.$
unlabeled · line 1405
$p_{ϵ} = \frac{p _{0}}{s _{ψ} ( g _{ϵ} , p _{0} )} .$
unlabeled · line 1413
$Π_{ϵ} u = s_{ψ} (u, p_{ϵ}) g_{ϵ} - s_{ψ} (u, g_{ϵ}) p_{ϵ} .$
unlabeled · line 1431
$σ_{R, ϵ} (W_{ψ} (u)) = σ_{ref} (W_{ψ} (S_{ϵ} u)) .$

An exact finite-plane squeeze

Displayed formulas in this passage

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