Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: How to perform the coherent measurement of a curved phase space by
  continuous isotropic measurement. I. Spin and the Kraus-operator geometry of
  $\mathrm{SL}(2,\mathbb{C})$   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
measurement's 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 $\mathrm{SU}(2)\backslash\mathrm{SL}(2,\mathbb{C})$. 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
$\mathrm{SU}(2)\subset\mathrm{SL}(2,\mathbb{C})$. 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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The categories of this paper are: patt-sol
11
The categories of this paper are: hep-lat
5
The categories of this paper are: gr-qc
1
The categories of this paper are: cs
31
The categories of this paper are: ao-sci
8
The categories of this paper are: hep-ex
34
The categories of this paper are: chem-ph
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The categories of this paper are: plasm-ph
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The categories of this paper are: nucl-th
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The categories of this paper are: q-fin
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The categories of this paper are: funct-an
3
The categories of this paper are: physics
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The categories of this paper are: q-alg
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The categories of this paper are: acc-phys
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The categories of this paper are: astro-ph
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The categories of this paper are: supr-con