Manuscript IV@1 · IV.tex · theorem · 351cb2d7

Theorem: Model 2 record: spectral removal at fixed M

Model 2: spectral removal at fixed number cutoff › Witness, calibration, and record · lines 2044–2108 · proof included or linked

IV@1#thm:model2

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Use the preceding detector and states, and fix $a, λ \neq = 0$ . Set

d = ∣ λa ∣ d_{*},

Q_{s} = 2 ∥ u_{0, f_{*}} ∥ ∥ v_{*} ∥^{2} + 1 + ∣ λ ∣ C_{B} μ_{s}^{2} + 5 μ_{s} + 4, S_{2} = Q_{0}^{2} + Q_{a}^{2} .

Choose, independently of $R, M$ ,

θ = π /2, 0 < τ \leq d / S_{2}, c_{2} = τ d /4.

For every fixed finite $M \geq 2$ , the Model 2 instruments satisfy

Δ_{R, M} := ∣ p^{(2)}_{+, R, M} (Ψ_{a}) - p^{(2)}_{+, R, M} (Ψ_{0}) ∣ \geq c_{2} - e_{M} - e_{R} (M),

where

e_{M} = \frac{τ ∣ λ ∣}{2} E_{M}, e_{R} (M) = ∣ τ λ ∣ δ_{M} (R) .

At that fixed $M$ , the generators, unitaries, and binary probabilities converge in operator norm, operator norm, and uniformly over states, respectively, as $R \to \infty$ . For all sufficiently large finite $M$ , the limiting contrast is at least $c_{2} /2$ .

At $λ = 0$ , the two probabilities agree exactly for every $R, M, τ, θ$ , including the fixed- $M$ spectral limit. No number-cutoff instrument limit is asserted.

Proof. At $R = \infty$ , the absolute generator mean difference is at least $d - ∣ λ ∣ E_{M}$ . The free scalar expectations cancel. The number bounds and Poisson moments give

A^{(2)}_{R, M} (λ) Ψ_{s} \leq Q_{s}

uniformly in $R, M$ . Lemma lem:Ramsey yields

Δ_{\infty, M} \geq \frac{τ}{2} (d - ∣ λ ∣ E_{M}) - \frac{τ ^{2} S _{2}}{4} \geq c_{2} - e_{M} .

The generator norm difference is $∣ λ ∣ δ_{M} (R)$ . The unitary difference is at most $∣ τ λ ∣ δ_{M} (R)$ ; each probability costs half this amount, so the contrast costs at most $e_{R} (M)$ . This proves all convergence and contrast statements. Since $E_{M} \to 0$ , one may fix a finite $M$ with $e_{M} \leq c_{2} /2$ .

At $λ = 0$ ,

e^{i τ A^{(2)}_{R, M} (0)} = I_{g} \otimes e^{i τ F_{0, M}} .

The scalar preparations coincide, proving the exact control.

Displayed formulas in this passage

unlabeled · line 2048
$d = ∣ λa ∣ d_{*},$
unlabeled · line 2051
$Q_{s} = 2 ∥ u_{0, f_{*}} ∥ ∥ v_{*} ∥^{2} + 1 + ∣ λ ∣ C_{B} μ_{s}^{2} + 5 μ_{s} + 4, S_{2} = Q_{0}^{2} + Q_{a}^{2} .$
unlabeled · line 2057
$θ = π /2, 0 < τ \leq d / S_{2}, c_{2} = τ d /4.$
unlabeled · line 2061
$Δ_{R, M} := ∣ p^{(2)}_{+, R, M} (Ψ_{a}) - p^{(2)}_{+, R, M} (Ψ_{0}) ∣ \geq c_{2} - e_{M} - e_{R} (M),$
unlabeled · line 2067
$e_{M} = \frac{τ ∣ λ ∣}{2} E_{M}, e_{R} (M) = ∣ τ λ ∣ δ_{M} (R) .$
unlabeled · line 2086
$A^{(2)}_{R, M} (λ) Ψ_{s} \leq Q_{s}$
unlabeled · line 2090
$Δ_{\infty, M} \geq \frac{τ}{2} (d - ∣ λ ∣ E_{M}) - \frac{τ ^{2} S _{2}}{4} \geq c_{2} - e_{M} .$
unlabeled · line 2103
$e^{i τ A^{(2)}_{R, M} (0)} = I_{g} \otimes e^{i τ F_{0, M}} .$

Theorem: Model 2 record: spectral removal at fixed M

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