TFPT scientific source excerpt
Original repository path: documents/archive--tfpt-45--source_extracts--technical_companion_source/sections/007-section-yukawa-kernels-and-positivity-lemmas.tex
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Inclusive original lines: 268-296
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000268 | \subsection{Observer algebra and Born-type evaluation}
000269 | 
000270 | An observer is modeled as a stable pointer subalgebra
000271 | \[
000272 | \Aobs\subset\Arec.
000273 | \]
000274 | 
000275 | \begin{theorem}[Born evaluation on the stable pointer algebra]
000276 | \label{thm:born-pointer-algebra}
000277 | Let $\{P_i\}\subset \Aobs$ be a complete family of orthogonal projectors. Then for every state
000278 | $\rho$ the unique additive, positive, and phase-invariant evaluation rule on this projector family is
000279 | \[
000280 | p_i=\tr(\rho P_i).
000281 | \]
000282 | \end{theorem}
000283 | 
000284 | \begin{proof}
000285 | In finite-dimensional sectors this is the standard Born rule characterization. The extra TFPT input
000286 | is not a different probability law but the selection of the stable pointer algebra by the gapped
000287 | admissible sector.
000288 | \end{proof}
000289 | 
000290 | With these projectors one may write the measurement channel
000291 | \[
000292 | \mathcal M_O(\rho)=\sum_i(P_i\rho P_i)\otimes |i\rangle\langle i|.
000293 | \]
000294 | The theorem is therefore a constrained appendix-level evaluation statement on admissible sectors,
000295 | not a new claim about measurement in arbitrary settings.
000296 |