The incompatibility between quantum measurements (as mathematically represented by positive operator-valued measures, i.e., POVMs) is a key feature of quantum mechanics and is intrinsically related to the noncommutativity of operators. For both theoretical and practical considerations, it is desirable to quantify the degree of incompatibility between quantum measurements, and considerable effort has been devoted to this issue. In this paper, we provide a novel approach to measurement incompatibility by exploiting the Lüders channels derived from POVMs and employing the measurement disturbance to quantify incompatibility. This is achieved by constructing an approximately joint measurement for a pair of POVMs, which is an enlarged POVM with the correct marginal property for one of the two POVMs but not necessarily for the other. The degree of failure of the marginal property for the other POVM is a kind of measurement disturbance and can be naturally interpreted as a quantifier of the incompatibility between the two POVMs. We reveal basic properties of this quantifier of measurement incompatibility, identify its maximal value in some cases, compare it with several popular measures in the literature, and illustrate it with some typical examples. Some related open issues are also discussed.
For any quantum observable (positive operator valued measure (POVM)), we show that its compatible quantum instruments can be identified with certain channels and can be seen as combinations of Lüders operations and channels. We prove that a POVM is rank-1 exactly when its compatible instruments are nuclear and their associate channels are entanglement-breaking. Any instrument can be maximally refined into a rank-1 instrument. We present a characterization for (minimal) pure realizations of instruments and characterize completely the minimal pure measurement models of POVMs. The standard model of quantum measurement theory is generalized for arbitrary observables. Finally, we study post measurement states.
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 SL(2,C)≅Spin(3,C) , a complex version of the usual unitary operators that represent elements of SU(2)≅Spin(3,R) . Consequently, the associated POVM elements represent points in the symmetric space SU(2)∖SL(2,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 SL(2,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 SU(2) ⊂ SL(2,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 SU(2) ⊂ SL(2,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.
This thesis presents a comprehensive study of phase-space measurements, decoherence, and the emergence of classicality in quantum systems. Motivated by limitations of position-based localisation in standard decoherence models, we develop a phase-space decoherence formalism using coherent-state positive operator-valued measures (POVMs). These generalised measurements, constructed from overcomplete sets of coherent states, provide a natural and physically motivated framework for modelling coarse-grained quantum observations. We develop an iterative measurement scheme for phase-space decoherence in systems with Heisenberg-Weyl,
SU(2),
SU(1,1), and
SU(N) symmetries. In each case, repeated weak measurements gradually suppress quantum interference, driving the system toward a maximally mixed state. A universal structure is shown to emerge: a single iteration maps the Wigner function to the Husimi
Q-function, removing negativity, while full decoherence maps the singular Glauber-Sudarshan
P-function to the
Q-function, eliminating all singularities. This establishes a group-independent mechanism by which phase-space measurements induce classicality. We compare this model to Lindblad-type dynamics, which describe open quantum systems via completely positive, trace-preserving semigroups. For Heisenberg-Weyl and tomographic
SU(N) systems, the measurement channel coincides exactly with isotropic Lindblad evolution. In
SU(2) and
SU(1,1) cases, the correspondence is only partial, with exact agreement in asymptotic regimes. A universal decoherence timescale,
t∗, emerges beyond which quasi-probability distributions become strictly positive. We then construct a stochastic unravelling based on Brownian motion on phase space, offering a geometric alternative to standard stochastic Schr\"odinger equations. This reveals that decoherence occurs without GKSL-type dynamics in noncompact systems. Such unravellings preserve the intrinsic phase-space symmetry and provide a natural setting for studying semiclassical limits and quantum-to-classical transitions. Finally, we extend the framework to relativistic quantum systems by defining a covariant coherent-state POVM on the future tube and constructing a Bargmann transform to the forward light cone. Preliminary results on relativistic phase-space decoherence are presented, along with possible directions for future research.