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

Christopher S. Jackson1,2 and Carlton M. Caves2

1Quantum Algorithms and Applications Collaboratory, Sandia National Laboratories, Livermore, CA 94550, USA
2Center for Quantum Information and Control, University of New Mexico, Albuquerque, NM 87131

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Abstract

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 POVM, placing spherical phase space at the boundary of the fundamental Lie group $\mathrm{SL}(2,{\mathbb C})$ in an operationally meaningful way. Two basic mathematical tools are used to analyze the evolving Kraus operators, the Maurer-Cartan form, modified for stochastic applications, and the Cartan, decomposition associated with the symmetric pair $\mathrm{SU}(2)$ ⊂ $\mathrm{SL}(2,{\mathbb C})$. Informed by these tools, the three schochastic techniques are applied directly to the Kraus operators in a representation-independent – and therefore geometric – way (independent of any spectral information about the spin components).
The Kraus-operator-centric, geometric treatment applies not just to $\mathrm{SU}(2)$ ⊂ $\mathrm{SL}(2,{\mathbb C})$, but also to any compact semisimple Lie group and its complexification. The POVM associated with the continuous isotropic measurement of Lie-group generators thus corresponds to a type-IV globally Riemannian symmetric space and limits to the POVM of generalized coherent states. This generalization is the focus of a sequel to this article.

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[2] Christopher S. Jackson and Carlton M. Caves, "Simultaneous Measurements of Noncommuting Observables: Positive Transformations and Instrumental Lie Groups", Entropy 25 9, 1254 (2023).

[3] Tathagata Karmakar, Philippe Lewalle, and Andrew N. Jordan, "Stochastic Path-Integral Analysis of the Continuously Monitored Quantum Harmonic Oscillator", PRX Quantum 3 1, 010327 (2022).

[4] Christopher S. Jackson and Carlton M. Caves, "Simultaneous Momentum and Position Measurement and the Instrumental Weyl-Heisenberg Group", Entropy 25 8, 1221 (2023).

[5] Christopher S. Jackson, "The photodetector, the heterodyne instrument, and the principle of instrument autonomy", arXiv:2210.11100, (2022).

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