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Foundations of Physics

On the simultaneous measurement of spin components using spin-1/2 meters: Naimark embedding and projections

Robert Y. Levine, Robert R. Tucci

Published Feb 1, 1989• 10 citations
Publisher
DOI
10.1007/bf00734524

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In analyzing two-electron systems, the interactions of interest often include the spin-spin operator S→1×S→2) and the spin-orbit operator L→·S→. When these operators act on entangled or indistinguishable particles, their measurement and physical interpretation may extend beyond the standard projective framework. This tutorial introduces the algebraic structure of spin interactions in two electron quantum systems and establishes its conceptual and mathematical connection with \emph{Naimark's Extension Theorem}. Through explicit examples for two-electron systems, we demonstrate how spin operators arise in reduced Hilbert spaces, and how \emph{Naimark's theorem} provides a formal framework for extending these to projective measurements in enlarged spaces. The application of \emph{Naimark's Extension Theorem} in deriving their matrix elements opens up a window into the structure of quantum measurements in such composite systems.

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In analyzing two-electron systems, the interactions of interest often include the spin-spin operator S→1×S→2) and the spin-orbit operator L→·S→. When these operators act on entangled or indistinguishable particles, their measurement and physical interpretation may extend beyond the standard projective framework. This tutorial introduces the algebraic structure of spin interactions in two electron quantum systems and establishes its conceptual and mathematical connection with \emph{Naimark's Extension Theorem}. Through explicit examples for two-electron systems, we demonstrate how spin operators arise in reduced Hilbert spaces, and how \emph{Naimark's theorem} provides a formal framework for extending these to projective measurements in enlarged spaces. The application of \emph{Naimark's Extension Theorem} in deriving their matrix elements opens up a window into the structure of quantum measurements in such composite systems.