Manuscript Em04-A@2 · Em04-A-v2.tex · theorem · f5b28abc

Theorem: Compressed-position POVM

Compressed oscillator position measurements › Exact objects and state approximation · lines 1052–1072 · proof included or linked

Em04-A@2#thm:compression

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The compressed effects form a countably additive normalized POVM:

E_{d} (B) \geq 0, E_{d} (R) = I_{d} .

Their sum over a countable measurable partition is $I_{d}$ . They are generally not projections and are not generally the spectral effects of $P_{d} X_{osc} P_{d}$ .

Proof. Positivity follows by testing on vectors in $H_{d}$ . Compression preserves countable additivity of matrix elements, and $P_{d} E (R) P_{d} = P_{d}$ . For $d = 1$ , the vacuum position density is $π^{- 1/2} e^{- x^{2}}$ , giving $E_{1} ((- \infty, 0]) = I_{1} /2$ , not a projection. The compressed position operator is zero, whose spectral effect for this set is instead $I_{1}$ .

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$E_{d} (B) \geq 0, E_{d} (R) = I_{d} .$

Theorem: Compressed-position POVM

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