Manuscript Psy02-A@1 · Psy02-A.tex · proposition · d92ce402

Proposition: Classical additive-noise POVM

Input, preparation, and measurement budgets › An additive-noise POVM and its exact cross term · lines 1482–1514 · proof included or linked

Psy02-A@1#prop:noise, Psy02-A@1#eq:noise-exact

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The operator-valued measure

F_{λ} (S) = \int_{R} λ (S - x) E_{B} (d x)

is a POVM. Its outcome law in a state $σ$ is the convolution of the spectral law of $B$ with $λ$ .

For finite second moments and independent $X, Z$ ,

E (X + Z - o)^{2} = E (X - o)^{2} + E Z^{2} + 2 (E X - o) E Z .

If the noise is centered, the last term vanishes. Without independence or a separately proved zero cross term, only the Minkowski allowance is asserted.

Proof. Positivity, normalization, and countable additivity follow from those of $λ$ and the spectral measure, using weak operator integration. The convolution statement follows by taking state expectations. Expand the square and use independence for the cross term.

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Psy02-A@1#eq:noise-exact · line 1495
$E (X + Z - o)^{2} = E (X - o)^{2} + E Z^{2} + 2 (E X - o) E Z .$
unlabeled · line 1485
$F_{λ} (S) = \int_{R} λ (S - x) E_{B} (d x)$

Proposition: Classical additive-noise POVM

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