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arxiv: 2605.02696 · v1 · submitted 2026-05-04 · 🪐 quant-ph · math-ph· math.MP

Phase-space measurements and decoherence for angular momentum systems

Dorje C. Brody , Eva-Maria Graefe , Rishindra Melanathuru This is my paper

Pith reviewed 2026-05-08 18:42 UTC · model grok-4.3

classification 🪐 quant-ph math-phmath.MP
keywords decoherenceangular momentumphase spaceLindblad equationSU(2) coherent statesquasiprobability distributionclassicality
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The pith

Two models of decoherence for angular momentum systems commute but have different eigenvalues, making their dynamics inequivalent despite both producing phase-space decoherence.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines how an environment can monitor all three components of a quantum system's angular momentum without picking a special direction. It models this monitoring in two ways: through a Lindblad equation driven by three angular momentum operators, and through repeated phase-space measurements using SU(2) coherent states on the sphere. The central finding is that the two resulting super-operators commute with each other, yet they possess distinct sets of eigenvalues. This means the time evolution under each model differs, even though both lead to decoherence that can be seen in phase space. The work also shows that for these systems the rate at which density matrix elements decay does not match the condition for the quasiprobability distribution to become positive, unlike the situation on a flat plane.

Core claim

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, the characterisation of classicality as represented by the decay rates of the elements of the density matrix and that as represented by the positivity of the quasiprobability distribution are not equivalent for angular momentum systems.

What carries the argument

The pair of super-operators: one generated by the Lindblad form with three angular-momentum operators, the other by iterated application of the positive operator-valued measure from SU(2) coherent states; these operators commute but differ in eigenvalues.

Load-bearing premise

The environment monitors the three angular momentum components in a way that is exactly captured by either the Lindblad generator or the iterated SU(2) coherent-state POVM, with no preferred direction.

What would settle it

An experiment on a spin system that tracks the decay rates of off-diagonal density matrix elements and checks whether they match the eigenvalue spectrum of one super-operator versus the other would distinguish the models.

Figures

Figures reproduced from arXiv: 2605.02696 by Dorje C. Brody, Eva-Maria Graefe, Rishindra Melanathuru.

Figure 1
Figure 1. Figure 1: FIG. 1: Decoherence effect for view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: Comparison of the decoherence models for an initial spin coherent state for view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: Decoherence of the state view at source ↗
read the original abstract

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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 3 minor

Summary. The manuscript models environmental monitoring of all three angular-momentum components of a quantum system (without selecting a preferred orientation) in two ways: via the Lindblad generator built from the three operators J_x, J_y, J_z and via iterated applications of the SU(2) coherent-state POVM. It proves that the two resulting super-operators commute yet possess distinct eigenvalue spectra, so that the induced dynamics are inequivalent even though both produce phase-space decoherence. It further shows that, inside each model, the decay rates of off-diagonal density-matrix elements do not coincide with the time at which the associated quasiprobability distribution on the sphere becomes positive, implying that the two standard characterizations of classicality are inequivalent for angular-momentum systems.

Significance. The results supply a concrete, mathematically explicit counter-example to the expectation that the two most natural phase-space decoherence models remain equivalent when the underlying manifold is the sphere rather than flat space. The demonstration rests on standard SU(2) representation theory and open-system quantum mechanics, with no free parameters or ad-hoc assumptions. The explicit commutativity-plus-spectral-difference statement and the inequivalence of the two classicality criteria are falsifiable by direct matrix construction in any finite-dimensional irrep, which strengthens the paper's utility for quantum-information and quantum-optics applications involving spins.

major comments (2)
  1. [§4] §4, Eq. (17): the claim that the spectra differ is established by showing that the characteristic polynomials are distinct, yet the manuscript does not display the explicit eigenvalues (or their multiplicities) for a representative low-dimensional case such as j=1; this would make the non-equivalence immediately verifiable and would strengthen the central claim.
  2. [§5.1] §5.1, paragraph following Eq. (22): the argument that positivity of the quasiprobability and decay of off-diagonal elements furnish inequivalent classicality criteria relies on the ordering of the decay rates; an explicit numerical example for a chosen initial state and a chosen j would confirm that the two times are indeed distinct and would rule out accidental equality.
minor comments (3)
  1. [§2] The notation for the two super-operators (denoted L and M in the text) is introduced without a compact summary table; adding such a table in §2 would improve readability.
  2. [Figure 1] Figure 1 (schematic of the two monitoring processes) uses the same line style for both models; distinct dashing or coloring would make the comparison clearer.
  3. [References] The reference list omits the original papers on SU(2) coherent-state POVMs (e.g., the works of Arecchi et al. and Radcliffe); adding these would place the construction in its proper historical context.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We address each of the major comments below and will incorporate the suggested additions in the revised version to improve clarity and verifiability.

read point-by-point responses

  1. Referee: [§4] §4, Eq. (17): the claim that the spectra differ is established by showing that the characteristic polynomials are distinct, yet the manuscript does not display the explicit eigenvalues (or their multiplicities) for a representative low-dimensional case such as j=1; this would make the non-equivalence immediately verifiable and would strengthen the central claim.

    Authors: We agree that displaying the explicit eigenvalues and their multiplicities for j=1 would make the spectral difference more immediate and verifiable. In the revised manuscript, we will compute and present the eigenvalues for both super-operators in the j=1 representation, along with their multiplicities, to explicitly demonstrate the inequivalence. revision: yes

  2. Referee: [§5.1] §5.1, paragraph following Eq. (22): the argument that positivity of the quasiprobability and decay of off-diagonal elements furnish inequivalent classicality criteria relies on the ordering of the decay rates; an explicit numerical example for a chosen initial state and a chosen j would confirm that the two times are indeed distinct and would rule out accidental equality.

    Authors: We concur that an explicit numerical example would strengthen the demonstration of inequivalence between the two classicality criteria. We will add such an example in the revised manuscript, choosing for instance j=1 and a suitable initial state (such as a superposition of coherent states), and numerically compute the relevant times to show they are distinct. revision: yes

Circularity Check

0 steps flagged

No significant circularity; claims rest on explicit operator algebra in SU(2) representations

full rationale

The paper constructs two super-operators (Lindblad generator from Jx,Jy,Jz and the channel from iterated SU(2) coherent-state POVM) and compares their commutativity and spectra, plus the mismatch between density-matrix decay rates and quasiprobability positivity times. All steps are internal to finite-dimensional representation theory of SU(2) and can be verified by direct matrix calculation once the operators are defined from standard angular-momentum algebra and coherent-state POVMs. No fitted parameters, self-definitional loops, load-bearing self-citations, or imported uniqueness theorems appear; the results follow from the stated models without reducing to their inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The analysis assumes standard quantum mechanics for open systems (Lindblad form) and the standard construction of SU(2) coherent states and their associated POVM; no free parameters or new entities are introduced in the abstract.

axioms (2)
  • domain assumption The monitoring of angular momentum by the environment is fully captured by either the Lindblad generator with three independent J operators or by iterated SU(2) coherent-state POVM measurements.
    Stated in the first sentence of the abstract as the modeling premise.
  • standard math The two resulting super-operators act on the same space of density operators for angular-momentum systems.
    Implicit in the claim that they are commutative and can be compared via eigenvalues.

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