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

Theorem: Local source-sign probability certificate

A local scalar probe for the sine witness › The local probability theorem · lines 1923–1979 · proof included or linked

Em03-F@1#thm:local, Em03-F@1#eq:localcertificate

Use the same independently prepared probe for both source signs. For either sign,

∣ (ω_{s} \otimes σ_{L, ϵ}) Θ_{ϵ} (I \otimes O_{s i n, ϵ}) - ω_{s} (E_{s i n, l}) ∣ \leq ε_{loc},

where

ε_{loc} = min {1, \frac{1 - c _{ϵ}}{2} + H_{C_{src}} (l) (δ_{L} + d_{p} (L, ϵ))} .

In the small-coupling regime,

ε_{loc} \leq min {1, H_{C_{src}} (l) δ_{L} + (H_{C_{src}} (l) T_{L} + N_{L} /4) ϵ^{2}},

T_{L} = 2 L_{p} Λ^{2} ∥ v_{A} ∥_{X} .

The local probability gap is at least

max {0, Δ_{s i n} - 2 ε_{loc}} .

Proof. The exact induced effect is (eq:inducedsine). Removing its attenuation changes an expectation by at most $(1 - c_{ϵ}) /2$ . The packet and transfer estimates give

∥ k (B_{ϵ}) - l ∥_{U} \leq δ_{L} + d_{p} .

Proposition prop:sourceclass verifies the same moment class for both source states, so Lemma lem:weylcontinuity gives (eq:localcertificate). Equations (eq:localtransfer)–(eq:localnoise) give the simplified form. The gap follows by the reverse triangle inequality.

Displayed formulas in this passage

Em03-F@1#eq:localcertificate · line 1927
$∣ (ω_{s} \otimes σ_{L, ϵ}) Θ_{ϵ} (I \otimes O_{s i n, ϵ}) - ω_{s} (E_{s i n, l}) ∣ \leq ε_{loc},$
unlabeled · line 1939
$ε_{loc} = min {1, \frac{1 - c _{ϵ}}{2} + H_{C_{src}} (l) (δ_{L} + d_{p} (L, ϵ))} .$
unlabeled · line 1948
$ε_{loc} \leq min {1, H_{C_{src}} (l) δ_{L} + (H_{C_{src}} (l) T_{L} + N_{L} /4) ϵ^{2}},$
unlabeled · line 1955
$T_{L} = 2 L_{p} Λ^{2} ∥ v_{A} ∥_{X} .$
unlabeled · line 1959
$max {0, Δ_{s i n} - 2 ε_{loc}} .$
unlabeled · line 1969
$∥ k (B_{ϵ}) - l ∥_{U} \leq δ_{L} + d_{p} .$

Nearby in this section

section: The local probability theorem