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Phase-space approaches to decoherence and classicality in quantum theory

Rishindra Melanathuru

Published Oct 2, 2025• 0 citations
Publisher

Abstract

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, $\mathrm{SU}(2)$, $\mathrm{SU}(1,1)$, and $\mathrm{SU}(N)$ 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 $\mathrm{SU}(N)$ systems, the measurement channel coincides exactly with isotropic Lindblad evolution. In $\mathrm{SU}(2)$ and $\mathrm{SU}(1,1)$ 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.

DOI
10.25560/126994

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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)\mathrm{SU}(2)SU(2), SU(1,1)\mathrm{SU}(1,1)SU(1,1), and SU(N)\mathrm{SU}(N)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 QQQ-function, removing negativity, while full decoherence maps the singular Glauber-Sudarshan PPP-function to the QQQ-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)\mathrm{SU}(N)SU(N) systems, the measurement channel coincides exactly with isotropic Lindblad evolution. In SU(2)\mathrm{SU}(2)SU(2) and SU(1,1)\mathrm{SU}(1,1)SU(1,1) cases, the correspondence is only partial, with exact agreement in asymptotic regimes. A universal decoherence timescale, t∗t_*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.
The emergence of classical behavior in quantum theory is often ascribed to the interaction of a quantum system with its environment, which can be interpreted as environmental monitoring of the system. As a result, off-diagonal elements of the density matrix of the system are damped in the basis of a preferred observable, often taken to be the position, leading to the phenomenon of decoherence. This effect can be modeled dynamically in terms of a Lindblad equation driven by the position operator. Here the question of decoherence resulting from a monitoring of position and momentum, i.e., a phase-space measurement, by the environment is addressed. There is no standard quantum observable corresponding to the detection of phase-space points, which is forbidden by Heisenberg's uncertainty principle. This issue is addressed by use of a coherent-state-based positive operator-valued measure for modeling phase-space monitoring by the environment. In this scheme, decoherence in phase space implies the diagonalization of the density matrix in both position and momentum representations. This is shown to be linked to a Lindblad equation where position and momentum appear as two independent Lindblad operators.
The decoherence phenomenon arising from an environmental monitoring of the state of a quantum system, as opposed to monitoring of a preferred observable, is worked out in detail using two equivalent formulations, namely, repeated applications of universal tomographic measurements using positive operator-valued measures, and its continuous time unravelling from the Lindblad equation. The effect of decoherence is analysed by studying the evolution of Stratonovich-Weyl quasiprobability distributions on the state-space of the system. It is shown that decoherence makes an arbitrary-given quasiprobability distribution manifestly positive, thus modelling the emergence of classicality in some sense. The decoherence timescale, the minimum time that quasiprobability distributions of every initial state of the system become nonnegative, is shown to decrease in Hilbert-space dimension, and hence larger quantum systems decohere faster.
Understanding how classical physics emerges from quantum mechanics remains a central problem in the foundations of physics. Here we derive a classical limit from finite-resolution measurements, modeled by continuous coarse-grained POVMs. When the resolved phase-space area is large compared with Planck's constant, the accessible statistics of any quantum state admit an effective classical description: coarse-grained observables become approximately jointly measurable and the induced probability density is positive. We derive the exact evolution equation for this density and show that, in the strong coarse-graining regime, its non-Liouville corrections are suppressed up to an Ehrenfest time, resulting in classical Hamiltonian flow generated by a Hamiltonian smoothed over the measurement cell. When the Hamiltonian varies negligibly across such a cell, the smoothed Hamiltonian reduces to the classical Hamiltonian whose quantization produced the quantum dynamics, thereby closing the quantization--classical-limit loop. Repeated finite-resolution measurements then generate stochastic records confined, with high probability, to tubes around classical trajectories. Our results provide a unified operational framework for the quantum-to-classical transition in microscopic and macroscopic systems, and establish the consistency of the quantization--classical-limit cycle.
This work presents a structural reformulation of quantum dynamics based on a scalar coherence functional defined over a general state space. The framework distinguishes two mathematically natural classes of dynamics generated by such a functional:(i) gradient flows, which monotonically increase coherence and correspond to stabilization or selection processes, and(ii) antisymmetric flows, which conserve coherence and generate reversible dynamics. We show that standard unitary quantum mechanics arises as a special case when the state space is a complex Hilbert space, the coherence functional is chosen as the negative energy expectation, and the antisymmetric structure corresponds to the canonical complex structure. In this setting, the Schrödinger equation follows directly from the formalism, without postulating unitarity as an independent axiom. The framework is extended to open systems using the density-operator formalism and completely positive trace-preserving maps. Under standard assumptions (Markovianity, continuity, and semigroup structure), the reduced dynamics takes the Gorini–Kossakowski–Sudarshan–Lindblad form, with dissipative terms interpreted as coherence-stabilizing dynamics. A supporting lemma demonstrates exponential suppression of off-diagonal density matrix elements under dephasing Lindblad operators. Using the Wigner phase-space representation, we further show that conservative coherence dynamics reduces to classical Liouville transport in the semiclassical regime, with quantum corrections governed by the Moyal bracket expansion. This work does not propose a modification of quantum theory. Rather, it offers a structural reorganization of existing formalisms. All results remain fully compatible with standard quantum mechanics while providing a unified perspective on unitary evolution, decoherence, and the emergence of classical behavior. Related work. A companion technical note, Structural Decomposition of SRG Recursion into Flow, Phase, and Closure (Halldórsson & ChatGPT, 2025, Zenodo), develops a discrete dynamical-systems formulation of SRG-style recursion and connects it explicitly to Lemmas 2.5, 2.7, and 2.9 of the present work.
The monitoring of the three independent components of the angular momentum (or spin) of a quantum system by its environment that does not isolate any preferred orientation is modelled in two different ways. One describes the dynamics by the Lindblad equation generated by three independent angular momentum operators. The other uses iterated measurements of the ``phase-space'' point on the sphere in terms of the positive operator-valued measure generated by SU(2) coherent states. In contrast to the equivalent scenario on a flat phase space, these two models give rise to subtle differences. Specifically, it is shown that the two super-operators corresponding to the two decoherence models for angular momentum systems are commutative, but their eigenvalues are different. Hence although both models give rise to phase-space decoherence, their dynamical behaviours are not equivalent. In either model, we find that the characterisation of classicality as represented by the decay rates of the elements of the density matrix (i.e. decoherence) and that as represented by the positivity of the quasiprobability distribution are not equivalent for angular momentum systems.
In this thesis, we study phase-space approximations to quantum dynamics for both Hermitian and non-Hermitian quantum systems, motivated by the need to efficiently describe quantum effects in models that are analytically intractable but admit classical analogues offering insight into their complex behaviour. Such systems arise in diverse areas, include but are not limited to quantum optics, condensed matter, open quantum systems, and quantum technologies. We begin by exploring phase-space approximations for quantum systems governed by the Heisenberg-Weyl algebra. Building on established coherent-state methods, we develop a semiclassical Husimi approximation for non-Hermitian Hamiltonians. This approach reveals that the resulting dynamics comprise the transport of initial Husimi distributions along classical trajectories, overlaid with a time-dependent norm landscape reflecting non-unitary evolution. We demonstrate the effectiveness of this method through several examples, including damped Kerr oscillators and PTPTPT-symmetric models, providing a superior approximation for observables compared to standard coherent-state techniques. To generalise these ideas, we develop a unified framework for phase-space approximations that applies across a wide range of algebraic structures. By identifying a set of general criteria, we enable systematic application of the approximation scheme to any quantum system satisfying these conditions. This provides a cohesive theoretical foundation that connects previously distinct results under a common approach. We then apply this framework to both the su2\mathfrak{su}_2su2​ algebra and non-Lie algebraic bosonic conversion systems. For su2\mathfrak{su}_2su2​, we analyse two-mode Bose-Hubbard models with various Hamiltonians, demonstrating quantum-classical correspondence and exploring the effects of PTPTPT-symmetry. For bosonic conversion systems, we propose coherent states for these non-Lie algebraic structures and derive their associated phase-space geometry, characterised as quantum Kummer shapes with classical counterparts as orbifolds. Our results advance the understanding of quantum-classical correspondence in multi-boson systems beyond traditional Hermitian frameworks and offer practical tools for simulating quantum dynamics in experimentally relevant models.