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

A numerical form of the noise bound

Probe preparation, noise, and squeezing cost › A numerical form of the noise bound · lines 1472–1514

Em03-D@1#eq:noisebound

Choose $X_{g}$ containing the spatial support of $g_{red}$ , and set

b_{g} = B_{X_{g}} (g_{red}) .

For $z_{ϵ} \leq 1/2$ , put

b_{J} = 2 L_{R} Λ∥ v_{A} ∥_{X} .

Then

V_{ref} (g_{ϵ}, g_{ϵ}) \leq b_{g} / ϵ + ϵ b_{J} .

For $0 < ϵ \leq 1$ , define

N_{R} = (b_{g} + b_{J})^{2} .

It follows that

V_{σ_{R, ϵ}} (g_{ϵ}, g_{ϵ}) \leq N_{R} ϵ^{2}, \frac{1 - c _{ϵ}}{2} \leq \frac{N _{R} ϵ ^{2}}{4} .

The second bound uses $1 - e^{- u} \leq u$ .

The normalization of $p_{ϵ}$ is also quantitatively controlled. Put

b_{P} = V_{ref} (p_{0}, p_{0}) .

The denominator cannot vanish if

ϵ^{2} 2 b_{J} b_{P} < ∣ c_{0} ∣/2,

because the symplectic form is bounded by twice the ordinary coefficient norms, which are dominated by the reference norms.

Displayed formulas in this passage

Em03-D@1#eq:noisebound · line 1493
$V_{σ_{R, ϵ}} (g_{ϵ}, g_{ϵ}) \leq N_{R} ϵ^{2}, \frac{1 - c _{ϵ}}{2} \leq \frac{N _{R} ϵ ^{2}}{4} .$
unlabeled · line 1476
$b_{g} = B_{X_{g}} (g_{red}) .$
unlabeled · line 1480
$b_{J} = 2 L_{R} Λ∥ v_{A} ∥_{X} .$
unlabeled · line 1484
$V_{ref} (g_{ϵ}, g_{ϵ}) \leq b_{g} / ϵ + ϵ b_{J} .$
unlabeled · line 1489
$N_{R} = (b_{g} + b_{J})^{2} .$
unlabeled · line 1505
$b_{P} = V_{ref} (p_{0}, p_{0}) .$
unlabeled · line 1509
$ϵ^{2} 2 b_{J} b_{P} < ∣ c_{0} ∣/2,$