Manuscript Hawk05-C@1 · Hawk05-C.tex · proposition · ba0719f1

Proposition: Mean occupation underdetermines capacity

Greybody weighting: what is calculable and what is declared › Occupation and flux are not dimensions · lines 1246–1266 · proof included or linked

Hawk05-C@1#prop:occupation

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For $0 < u < 1$ , a bosonic number mean $u$ is compatible with both

(1 - u) ∣ 0 ⟩ ⟨ 0 ∣ + u ∣ 1 ⟩ ⟨ 1 ∣

and the geometric state

ρ_{u} = (1 - z) n = 0 \sum^{\infty} z^{n} ∣ n ⟩ ⟨ n ∣, z = \frac{u}{1 + u} .

They have different spectra and support dimensions. Therefore an occupation mean does not determine a Hilbert-space dimension or a Haar state ensemble.

Proof. Both states have mean $u$ , by a direct sum of their number probabilities. The first has rank two and the second has infinite rank. Equality of one number moment does not identify their states.

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unlabeled · line 1249
$(1 - u) ∣ 0 ⟩ ⟨ 0 ∣ + u ∣ 1 ⟩ ⟨ 1 ∣$
unlabeled · line 1253
$ρ_{u} = (1 - z) n = 0 \sum^{\infty} z^{n} ∣ n ⟩ ⟨ n ∣, z = \frac{u}{1 + u} .$

Proposition: Mean occupation underdetermines capacity

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