Manuscript En10-B@1 · En10-B.tex · lemma · a1105bad

Lemma: Exact record law

Likelihood, source sensitivity, and nuisance information › The derived four-outcome law · lines 597–659 · proof included or linked

En10-B@1#lem:law, En10-B@1#eq:likelihood, En10-B@1#eq:score, En10-B@1#eq:affine

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On input $τ_{θ}$ , the derived acquisition probabilities are

p_{00} (θ, η) p_{01} (θ, η) p_{10} (θ, η) p_{11} (θ, η) = γ u + h v + θ d, = h (u - v), = γ + (1 - 2 γ) u - h v - γ θ d, = h (1 - 2 u + v) - h θ d, p_{⊥} = 0.

All four nonfailure probabilities are strictly positive. For the complete-record score

f (r) = {1, 0, r = 00 or 10, r = 01, 11, ⊥,

its ideal mean is

q_{η} (θ) = a_{η} + b_{η} θ, a_{η} = γ + h u, b_{η} = h (1 - 2 u) > 0.

Proof. Since $K_{n}^{2} = I$ ,

V_{k} = cos (ϕ + ξ_{k}) I - i sin (ϕ + ξ_{k}) K_{n} .

Both $V_{k}$ and $V_{k}^{†}$ have squared matrix elements

(1 - t_{k} t_{k} t_{k} 1 - t_{k}) .

The diagonal interaction changes no occupation population.

Conditional on source component $00$ and label $k$ , the four probabilities after relaxation are

((1 - t_{k}) (1 - h t_{k}), h (t_{k} - t_{k}^{2}), t_{k} - h t_{k}^{2}, h t_{k}^{2}) .

Conditional on source component $11$ , they are

(γ t_{k} + h t_{k}^{2}, h (t_{k} - t_{k}^{2}), γ + (1 - 2 γ) t_{k} - h t_{k}^{2}, h (1 - t_{k})^{2}) .

The population update follows directly from the two Kraus operators $L_{ℓ}$ ; coherences do not contribute to these computational probabilities. Mixing with source weights $θ, 1 - θ$ and then weights $w_{k}$ proves (eq:likelihood). Every conditional entry is positive because $0 < t_{k} < 1/2$ and $h > 0$ . The averaged entries are therefore positive. Summing $p_{00} + p_{10}$ gives (eq:affine).

Displayed formulas in this passage

En10-B@1#eq:likelihood · line 600
$p_{00} (θ, η) p_{01} (θ, η) p_{10} (θ, η) p_{11} (θ, η) = γ u + h v + θ d, = h (u - v), = γ + (1 - 2 γ) u - h v - γ θ d, = h (1 - 2 u + v) - h θ d, p_{⊥} = 0.$
En10-B@1#eq:score · line 611
$f (r) = {1, 0, r = 00 or 10, r = 01, 11, ⊥,$
En10-B@1#eq:affine · line 620
$q_{η} (θ) = a_{η} + b_{η} θ, a_{η} = γ + h u, b_{η} = h (1 - 2 u) > 0.$
unlabeled · line 629
$V_{k} = cos (ϕ + ξ_{k}) I - i sin (ϕ + ξ_{k}) K_{n} .$
unlabeled · line 634
$(1 - t_{k} t_{k} t_{k} 1 - t_{k}) .$
unlabeled · line 641
$((1 - t_{k}) (1 - h t_{k}), h (t_{k} - t_{k}^{2}), t_{k} - h t_{k}^{2}, h t_{k}^{2}) .$
unlabeled · line 646
$(γ t_{k} + h t_{k}^{2}, h (t_{k} - t_{k}^{2}), γ + (1 - 2 γ) t_{k} - h t_{k}^{2}, h (1 - t_{k})^{2}) .$

Lemma: Exact record law

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