Manuscript Hawk02-E@1 · Hawk02-E.tex · proposition · 4841592b

Proposition: Preparation probability and energy distribution

Pure preparation and coherent diary loading › Source preparation · lines 581–611 · proof included or linked

Hawk02-E@1#prop:preparation

The state (eq:KM-prep) is well defined and has the same parity as $∣ B_{s} ⟩$ . Its normalization obeys

e^{- β M_{N}} \leq Z_{s} (β) \leq 1.

If $H_{S} = \sum_{E} E P_{E}$ , its energy probabilities are

p_{E} = Z_{s} (β)^{- 1} e^{- β E} ⟨ B_{s} ∣ P_{E} ∣ B_{s} ⟩ .

Moreover,

⟨ H_{S} ⟩_{Ω_{S}} = - \partial_{β} lo g Z_{s} (β), Var_{Ω_{S}} (H_{S}) = \partial_{β}^{2} lo g Z_{s} (β) .

Proof. The spectral bounds $0 \leq H_{S} \leq M_{N} I$ give the normalization inequalities and positivity of $Z_{s}$ . The quartic Hamiltonian is even, so its exponential preserves parity. The spectral decomposition gives $p_{E}$ . Differentiating the finite-dimensional spectral sum proves the mean and variance formulas.

Displayed formulas in this passage

unlabeled · line 585
$e^{- β M_{N}} \leq Z_{s} (β) \leq 1.$
unlabeled · line 589
$p_{E} = Z_{s} (β)^{- 1} e^{- β E} ⟨ B_{s} ∣ P_{E} ∣ B_{s} ⟩ .$
unlabeled · line 595
$⟨ H_{S} ⟩_{Ω_{S}} = - \partial_{β} lo g Z_{s} (β), Var_{Ω_{S}} (H_{S}) = \partial_{β}^{2} lo g Z_{s} (β) .$

Proposition: Preparation probability and energy distribution

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