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

Lemma: Weyl expectation comparison

The outgoing target and a controlled state class · lines 689–727 · proof included or linked

Em03-D@1#lem:Weylcomparison, Em03-D@1#eq:effectcomparison

For $ω \in S_{C}$ and weighted coefficients $k, l$ ,

∣ ω (W (k)) - ω (W (l)) ∣ \leq C ∥ k - l ∥_{U} + \frac{∣ s ( k , l ) ∣}{2} .

In particular,

ω (\frac{I + Re W ( k )}{2}) - ω (\frac{I + Re W ( l )}{2}) \leq \frac{C + ∥ l ∥ _{U}}{2} ∥ k - l ∥_{U} .

Proof. Put $d = k - l$ . The Weyl relation expresses $W (k)$ as $W (l) W (d)$ times a scalar phase of angle of magnitude $∣ s (k, l) ∣/2$ . Cauchy–Schwarz for the state and spectral calculus give

∣ ω (W (l) (W (d) - I)) ∣ \leq ω ((W (d) - I)^{*} (W (d) - I)) \leq ω (X (d)^{2}) .

The phase costs at most $∣ s (k, l) ∣/2$ . This proves the first inequality. Since

∣ s (k, l) ∣ = ∣ s (d, l) ∣ \leq 2∥ d ∥_{0} ∥ l ∥_{0} \leq 2∥ d ∥_{U} ∥ l ∥_{U},

taking real parts and dividing by two proves the second.

Displayed formulas in this passage

Em03-D@1#eq:effectcomparison · line 697
$ω (\frac{I + Re W ( k )}{2}) - ω (\frac{I + Re W ( l )}{2}) \leq \frac{C + ∥ l ∥ _{U}}{2} ∥ k - l ∥_{U} .$
unlabeled · line 692
$∣ ω (W (k)) - ω (W (l)) ∣ \leq C ∥ k - l ∥_{U} + \frac{∣ s ( k , l ) ∣}{2} .$
unlabeled · line 714
$∣ ω (W (l) (W (d) - I)) ∣ \leq ω ((W (d) - I)^{*} (W (d) - I)) \leq ω (X (d)^{2}) .$
unlabeled · line 721
$∣ s (k, l) ∣ = ∣ s (d, l) ∣ \leq 2∥ d ∥_{0} ∥ l ∥_{0} \leq 2∥ d ∥_{U} ∥ l ∥_{U},$

Lemma: Weyl expectation comparison

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