Recently it was reported that the spin-coherent state (SCS) positive-operator-valued measure (POVM) can be performed for any spin system by continuous isotropic measurement of the three total spin components [E. Shojaee, C. S. Jackson, C. A. Riofrio, A. Kalev, and I. H. Deutsch, Phys. Rev. Lett. 121, 130404 (2018)]. The outcome probability distribution of the SCS POVM for an input quantum state is the generalized $Q$-function, which is defined on the 2-sphere phase space of SCSs. This article develops the theoretical details of the continuous isotropic measurement and places it within the general context of applying curved-phase-space correspondences to quantum systems, indicating their experimental utility by explaining how to analyze this measurements performance. The analysis is in terms of the Kraus operators that develop over the course of a continuous isotropic measurement. The Kraus operators represent elements of the Lie group $mathrm{SL}(2,mathbb{C})$, a complex version of the usual unitary operators that represent elements of $mathrm{SU}(2)$. Consequently, the associated POVM elements represent points in the 3-hyperboloid $mathrm{SU}(2)backslashmathrm{SL}(2,mathbb{C})$. Three equivalent stochastic techniques, path integral, diffusion (Fokker-Planck) equation, and stochastic differential equations, are applied to show that the POVM quickly limits to the SCS POVM. We apply two basic mathematical tools to the Kraus operators, the Maurer-Cartan form, modified for stochastic applications, and the Cartan decomposition associated with the symmetric pair $mathrm{SU}(2)subsetmathrm{SL}(2,mathbb{C})$. Informed by these tools, the three stochastic techniques are applied directly to the Kraus operators in a representation independent, and thus geometric, way (independent of any spectral information about the spin components).
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