Manuscript Em03-F@1 · Em03-F-v1.tex · theorem · 0d4d148b

Theorem: Exact source-sign witness

Phase-sensitive witnesses and what number counts miss · lines 1112–1154 · proof included or linked

Em03-F@1#thm:witness, Em03-F@1#eq:shiftedcharacteristic, Em03-F@1#eq:singap

With the evolution sign fixed in (eq:UF),

ω_{s} (W (l)) = e^{- ∥ l ∥_{U}^{2} /2} e^{- i s g σ (k_{F}, l)} .

Hence

ω_{s} (E_{s i n, l}) = \frac{1}{2} [1 - s e^{- ∥ l ∥_{U}^{2} /2} sin (g σ (k_{F}, l))],

and the sign gap is

Δ_{s i n} = e^{- ∥ l ∥_{U}^{2} /2} ∣ sin (g σ (k_{F}, l)) ∣.

The cosine expectations are identical for both signs. If $l$ belongs to the selected outgoing packet algebra, $d_{h} \geq Δ_{s i n}$ .

Proof. The exact Weyl identity is

W (r)^{†} W (l) W (r) = e^{i σ (r, l)} W (l) .

Insert $r = - s g k_{F}$ , then the centered Gaussian characteristic function. Real and imaginary parts give the two effects. An effect probability difference is bounded by state trace distance, proving the last assertion.

Displayed formulas in this passage

Em03-F@1#eq:shiftedcharacteristic · line 1115
$ω_{s} (W (l)) = e^{- ∥ l ∥_{U}^{2} /2} e^{- i s g σ (k_{F}, l)} .$
Em03-F@1#eq:singap · line 1131
$Δ_{s i n} = e^{- ∥ l ∥_{U}^{2} /2} ∣ sin (g σ (k_{F}, l)) ∣.$
unlabeled · line 1124
$ω_{s} (E_{s i n, l}) = \frac{1}{2} [1 - s e^{- ∥ l ∥_{U}^{2} /2} sin (g σ (k_{F}, l))],$
unlabeled · line 1146
$W (r)^{†} W (l) W (r) = e^{i σ (r, l)} W (l) .$

Nearby in this section

section: Phase-sensitive witnesses and what number counts miss