How to perform the coherent measurement of a curved phase space by continuous isotropic measurement. I. Spin and the Kraus-operator geometry of $\mathrm{SL}(2,\mathbb{C})$

The generalized $Q$-function of a spin system can be considered the outcome probability distribution of a state subjected to a measurement represented by the spin-coherent-state (SCS) positive-operator-valued measure (POVM). As fundamental as the SCS POVM is to the 2-sphere phase-space representation of spin systems, it has only recently been reported that the SCS 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)]. This article develops the theoretical details of the continuous isotropic measurement and places it within the general context of curved-phase-space correspondences for quantum systems. The analysis is in terms of the Kraus operators that develop over the course of a continuous isotropic measurement. The Kraus operators of any spin $j$ are shown to represent elements of the Lie group $\mathrm{SL}(2,{\mathbb C})\cong\mathrm{Spin}(3,{\mathbb C})$, a complex version of the usual unitary operators that represent elements of $\mathrm{SU}(2)\cong\mathrm{Spin}(3,{\mathbb R})$. Consequently, the associated POVM elements represent points in the symmetric space $\mathrm{SU}(2)\backslash\mathrm{SL}(2,{\mathbb C})$, which can be recognized as the 3-hyperboloid. Three equivalent stochastic techniques, (Wiener) path integral, (Fokker-Planck) diffusion equation, and stochastic differential equations, are applied to show that the continuous isotropic POVM quickly limits to the SCS~\hbox{POVM}, placing spherical phase space at the boundary of the fundamental Lie group $\mathrm{SL}(2,{\mathbb C})$ in an operationally meaningful way. The Kraus-operator-centric analysis is representation independent -- and therefore geometric (independent of any spectral information about the spin components).