Manuscript Hawk01-B@1 · Hawk01-B.tex · proposition · cfa976f1

Proposition: Local deviation certificate

Hawking-mode calibration and compatibility limits · lines 1291–1321 · proof included or linked

Hawk01-B@1#prop:thermalcomparison, Hawk01-B@1#eq:Bernoullideviation, Hawk01-B@1#eq:thermalcutdistance, Hawk01-B@1#eq:fullthermaldeviation

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For

π_{g} (w) = (1 - w) ∣ vac ⟩ ⟨ vac ∣ + w I_{g}^{(1)} / g,

one has

D_{tr} (π_{g} (w), χ_{g} (q)) D_{tr} (χ_{g} (q), σ_{g} (q)) D_{tr} (π_{g} (w), σ_{g} (q)) = ∣ w - a_{g} (q) ∣, = θ_{g} (q), \leq min {1, ∣ w - a_{g} (q) ∣ + θ_{g} (q)} .

Thus the transfer block $R_{j}$ has the displayed bound with $g = d$ and $w = w_{j}$ .

Proof. The vacuum and uniform one-particle sectors are orthogonal, so their probability difference gives (eq:Bernoullideviation). The conditional low-occupation state and the removed tail occupy orthogonal sectors; renormalization gives trace distance equal to the removed probability. Apply the triangle inequality.

Displayed formulas in this passage

Hawk01-B@1#eq:Bernoullideviation · line 1299
$D_{tr} (π_{g} (w), χ_{g} (q)) D_{tr} (χ_{g} (q), σ_{g} (q)) D_{tr} (π_{g} (w), σ_{g} (q)) = ∣ w - a_{g} (q) ∣, = θ_{g} (q), \leq min {1, ∣ w - a_{g} (q) ∣ + θ_{g} (q)} .$
Hawk01-B@1#eq:thermalcutdistance · line 1299
$D_{tr} (π_{g} (w), χ_{g} (q)) D_{tr} (χ_{g} (q), σ_{g} (q)) D_{tr} (π_{g} (w), σ_{g} (q)) = ∣ w - a_{g} (q) ∣, = θ_{g} (q), \leq min {1, ∣ w - a_{g} (q) ∣ + θ_{g} (q)} .$
Hawk01-B@1#eq:fullthermaldeviation · line 1299
$D_{tr} (π_{g} (w), χ_{g} (q)) D_{tr} (χ_{g} (q), σ_{g} (q)) D_{tr} (π_{g} (w), σ_{g} (q)) = ∣ w - a_{g} (q) ∣, = θ_{g} (q), \leq min {1, ∣ w - a_{g} (q) ∣ + θ_{g} (q)} .$
unlabeled · line 1294
$π_{g} (w) = (1 - w) ∣ vac ⟩ ⟨ vac ∣ + w I_{g}^{(1)} / g,$

Proposition: Local deviation certificate

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