Manuscript Em03-H@8 · Em03-H-v8.tex · theorem · 91e9e3bd

Theorem: Departure and short-time estimates

Quench leakage, best compression, and departure · lines 673–704 · proof included or linked

Em03-H@8#thm:departure

p_{m a x} (u) = (I - P) e^{- i u N_{B}} V^{2} = 1 - λ_{m i n} (F (u)^{†} F (u)) .

For $u \geq 0$ ,

max (0, u L_{N} - u^{2} n^{2} /8)^{2} \leq p_{m a x} (u) \leq min (1, (u L_{N} + u^{2} n^{2} /8)^{2}),

∣ p_{m a x} (u) - u^{2} L_{N}^{2} ∣ \leq u^{3} n^{2} L_{N} /4 + u^{4} n^{4} /64.

For general $Ω$ , replace $u$ by $Ω u$ .

Proof. Retained probability is $∥ F (u) x ∥^{2}$ ; optimize its complement. Center at $A_{0} = N_{B} - n I /2$ , with norm $n /2$ . The scalar phase does not affect departure, and the integral Taylor remainder gives

e^{- i u A_{0}} - I + i u A_{0} \leq u^{2} n^{2} /8.

The off-code amplitude is $- i u (I - P) N_{B} V + R_{u}$ , with the displayed remainder bound. Triangle inequalities give the first bounds; the squared-norm difference is at most $2 u L_{N} ∥ R_{u} ∥ + ∥ R_{u} ∥^{2}$ .

Displayed formulas in this passage

unlabeled · line 675
$p_{m a x} (u) = (I - P) e^{- i u N_{B}} V^{2} = 1 - λ_{m i n} (F (u)^{†} F (u)) .$
unlabeled · line 681
$max (0, u L_{N} - u^{2} n^{2} /8)^{2} \leq p_{m a x} (u) \leq min (1, (u L_{N} + u^{2} n^{2} /8)^{2}),$
unlabeled · line 686
$∣ p_{m a x} (u) - u^{2} L_{N}^{2} ∣ \leq u^{3} n^{2} L_{N} /4 + u^{4} n^{4} /64.$
unlabeled · line 697
$e^{- i u A_{0}} - I + i u A_{0} \leq u^{2} n^{2} /8.$

Theorem: Departure and short-time estimates

Displayed formulas in this passage

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