Manuscript Hawk07-A@1 · Hawk07-A.tex · theorem · f9c0f5da

Theorem: One threshold for all later retained cuts

The Haar decoupling calculation · lines 861–895 · proof included or linked

Hawk07-A@1#thm:threshold, Hawk07-A@1#eq:threshold, Hawk07-A@1#eq:simplethreshold

Fix $0 < p < 1$ , $ζ > 0$ , and a cutoff $K$ . If a retained cut $k_{*} \leq K$ satisfies

K (r_{k_{*}}, b_{k_{*}}) \leq p ζ^{2},

then, with Haar probability at least $1 - p$ , the same sampled unitary satisfies, for every $k_{*} \leq k \leq K$ ,

d_{⋄} (C_{k}^{U}, T_{σ_{B_{k}}^{U}}) \leq ζ, ε_{rec} (N_{k}^{U}) \leq min {a_{d}, 2 ζ} .

A sufficient threshold is

r_{k_{*}} \geq \frac{d ^{2} f}{2 ζ p} .

Proof. Markov's inequality gives

P {z_{k_{*}} (U) > ζ} \leq K (r_{k_{*}}, b_{k_{*}}) / ζ^{2} \leq p .

Proposition prop:nested propagates this single event to all later retained cuts. Lemma lem:KSW supplies recovery. Equation (eq:simplethreshold) follows from (eq:K).

Displayed formulas in this passage

Hawk07-A@1#eq:threshold · line 865
$K (r_{k_{*}}, b_{k_{*}}) \leq p ζ^{2},$
Hawk07-A@1#eq:simplethreshold · line 877
$r_{k_{*}} \geq \frac{d ^{2} f}{2 ζ p} .$
unlabeled · line 871
$d_{⋄} (C_{k}^{U}, T_{σ_{B_{k}}^{U}}) \leq ζ, ε_{rec} (N_{k}^{U}) \leq min {a_{d}, 2 ζ} .$
unlabeled · line 886
$P {z_{k_{*}} (U) > ζ} \leq K (r_{k_{*}}, b_{k_{*}}) / ζ^{2} \leq p .$

Theorem: One threshold for all later retained cuts

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