Measurement theory in local quantum physics (Q2786614)

The paper establishes a modality for introducing measurement theory in the field of local quantum physics. The main tool used is the ``completely positively'' (CP) instrument, defined three decades ago in [the second author, ``Quantum measuring processes of continuous observables'', J. Math. Phys. 25, 79--87 (1984; \url{doi:10.1063/1.526000})] for quantum mechanic field, extended here to general von Neumann algebras, and applied in the case of quantum systems with infinite degrees of freedom. The authors define a special property of CP instruments, called ``normal extension property'' (NEP) and prove that NEP for CP instruments is equivalent with the existence of the corresponding measuring processes. More precisely, the next results are obtained for a von Neumann algebra \(\mathcal M\) and a measurable space \(\mathcal S\):NEWLINE{\parindent=6mm NEWLINE\begin{itemize} \item[(1)] There is a one-to-one correspondence between statistical equivalent classes of measuring processes and CP instruments with NEP (Theorem 3.6);NEWLINE\item [(2)] If \(\mathrm{CPInst}({\mathcal M},{\mathcal S})\) is the set of CP instruments, and \(\mathrm{CPInst}_{NE}({\mathcal M},{\mathcal S})\) (\(\mathrm{CPInst}_{AN}({\mathcal M},{\mathcal S})\)) is the set of instruments with the NEP (Approximately normal extension property -- ANEP respectively), thenNEWLINENEWLINE\end{itemize}} NEWLINE{\parindent=12mmNEWLINE\begin{itemize} \item[(2.a)] \(\mathrm{CPInst}_{NE}({\mathcal M},{\mathcal S})=\mathrm{CPInst}_{AN}({\mathcal M},{\mathcal S})=\mathrm{CPInst}({\mathcal M},{\mathcal S})\) if \(\mathcal M\) is atomic,NEWLINE\item [(2.b)] \(\mathrm{PInst}_{AN}({\mathcal M},{\mathcal S})=\mathrm{CPInst}({\mathcal M},{\mathcal S})\) if \(\mathcal M\) is injective (Theorem 4.4).NEWLINENEWLINE\end{itemize}} NEWLINE{\parindent=6mm NEWLINE\begin{itemize} \item[(3)] For \(\mathcal M\) non-atomic but injective, the inclusion \(\mathrm{CPInst}_{NE}({\mathcal M},{\mathcal S})\subset \mathrm{CPInst}_{AN}({\mathcal M},{\mathcal S})\) is strict (Section 5).NEWLINE\item [(4)] For any CP instrument, the NEP is equivalent to the existence of a strongly measurable family of aposteriori states, for every normal state (Corollary 5.6).NEWLINE\item [(5)] A weakly repeatable CP instrument has NEP if and only if it is discrete (an extension of a similar result of \textit{E. B. Davies} and \textit{J. T. Lewis} [Commun. Math. Phys. 17, 239--260 (1970; Zbl 0194.58304)]) (Theorem 5.10).NEWLINE\item [(6)] Every CP instrument defined on a local algebra represents a local instrument measurement within arbitrary error limits, and (with some restrictions) vice versa. Therefore, an instrument on a local algebra can be extended to a local instrument on a global algebra if and only if it is a CP instrument (Theorems 6.1 and 6.2). NEWLINENEWLINE\end{itemize}} NEWLINEFinally, the authors express their hope that ``this paper would be helpful for developing measurement theory in local quantum physics''.