Which Coherence Decoheres? Basis-Dependent Decoherence Rates in Symmetry-Broken Collective Spin Systems

Stavros Mouslopoulos

arxiv: 2605.00952 · v1 · submitted 2026-05-01 · 🪐 quant-ph · hep-th

Which Coherence Decoheres? Basis-Dependent Decoherence Rates in Symmetry-Broken Collective Spin Systems

Stavros Mouslopoulos This is my paper

Pith reviewed 2026-05-09 20:01 UTC · model grok-4.3

classification 🪐 quant-ph hep-th

keywords collective spin systemsdecoherence ratesZ2 symmetrypointer statesenergy eigenstatesLindblad dynamicsLipkin-Meshkov-Glick modelparity selection rules

0 comments

The pith

Parity symmetry in collective spin systems protects energy eigenstates against decoherence more than localized pointer states.

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

This paper shows that in the ordered phase of Z2-symmetric collective spin systems, two different bases for the same coherent superposition yield different Lindblad dephasing rates. Localized pointer states decohere faster because their rate includes a cross-term from nonzero diagonal Jz elements, while parity forces those elements to zero in the energy basis. The resulting protection factors are 2.42 in a mean-field approximation and 1.86 exactly. As system size grows to infinity, the energy doublet becomes degenerate, the secular approximation fails, and the rates converge. These findings are verified through exact diagonalization of the Lipkin-Meshkov-Glick model.

Core claim

The discrepancy in decoherence rates between localized pointer states and energy eigenstates arises solely from the Z2 parity of the Lindblad jump operator, which sets the expectation value of Jz to zero in each energy eigenstate. This eliminates the cross-term that doubles the dephasing rate for pointer states, leading to distinct protection factors η_MF ≈ 2.42 and η_exact ≈ 1.86, which both approach unity in the thermodynamic limit where rates converge.

What carries the argument

The Z2 parity symmetry of the Lindblad jump operator, which enforces ⟨Ei|Jz|Ei⟩ = 0 and removes the cross-term in the dephasing rate formula.

If this is right

Decoherence rates for the two bases become identical in the thermodynamic limit as the secular approximation fails and the doublet degenerates.
The three-regime structure (finite-N discrepancy, quantum-critical crossover, convergence) is confirmed by exact diagonalization in the Lipkin-Meshkov-Glick model.
The algebraic origin of the discrepancy holds for any Z2-symmetric collective spin system in the ordered phase.
The mean-field protection factor overestimates the exact physical ratio of pointer-state to eigenstate decay rates.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Choosing energy eigenstates rather than localized pointer states could provide built-in protection against decoherence in symmetric open quantum systems used for information processing.
The basis dependence may influence how macroscopic coherence is defined and observed in symmetry-broken phases of many-body systems.
Similar parity-driven rate differences could appear in other open quantum models with discrete symmetries beyond spin systems.
Experiments measuring decay of superpositions in large trapped-ion or Rydberg-atom ensembles could test the predicted convergence at large N.

Load-bearing premise

The Lindblad jump operator respects the Z2 parity of the system and the dynamics remain within the ordered phase where the two bases are well-defined and distinct.

What would settle it

Exact computation of dephasing rates in the LMG model at moderate N showing no difference between bases, or equal rates when parity is artificially broken.

Figures

Figures reproduced from arXiv: 2605.00952 by Stavros Mouslopoulos.

**Figure 1.** Figure 1: shows ηMF versus N at Γ/J = 0.95. 200 400 600 800 1,000 1,200 1,400 1,600 1,800 2,000 1 1.5 2 2.5 ∆ E = kBT (N = 370) N ηMF = Gloc/G01 Exact diagonalisation ηMF = 2 (N → ∞) view at source ↗

**Figure 2.** Figure 2: shows ηMF versus Γ/J at N = 370. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.5 1 1.5 2 2.5 Γ/J=0.95 ηMF=2.35 peak: Γ/J=0.96 ηMF=2.42 Γ/J ηMF = Gloc/G01 N = 370, exact diag. ηMF = 2 (N → ∞) view at source ↗

**Figure 3.** Figure 3: shows all three rates normalised by the common scale (Nm∗/2)2 , making the convergence visually explicit. 200 400 600 800 1,000 1,200 1,400 1,600 1,800 2,000 1 1.5 2 2.5 Secular window → 2 → 1 N Rate factor / (Nm∗/2) 2 Gloc/(Nm∗/2)2 = 2 2J 2 01/(Nm∗/2)2 (restricted doublet eigenvalue) G01/(Nm∗/2)2 (secular rate) (Nm∗/2)2/(Nm∗/2)2 = 1 view at source ↗

read the original abstract

In the ordered phase of a $\mathbb{Z}_2$-symmetric collective spin system, two natural bases -- localised pointer states $\{|P\rangle,|R\rangle\}$ and energy eigenstates $\{|E_0\rangle,|E_1\rangle\}$ -- yield Lindblad dephasing rates that differ by a factor approaching $2$ as $N\to\infty$ and reaching $2.42$ near the quantum-critical crossover. The discrepancy has a single algebraic origin: parity forces $\langle E_i|\hat{J}_z|E_i\rangle=0$ exactly, eliminating the cross-term that doubles the localised-state rate. Two distinct protection factors are identified: $\eta_{\rm MF}=(Nm_*)^2/(2G_{01})\approx2.42$, where $m_*$ is the order parameter and $G_{01}=\frac{1}{2}(\langle E_0|\hat{J}_z^2|E_0\rangle+\langle E_1|\hat{J}_z^2|E_1\rangle)$ (advantage over the classical mean-field estimate), and $\eta_{\rm exact}=(G_{01}+J_{01}^2)/G_{01}\approx1.86$, where $J_{01}=\langle E_0|\hat{J}_z|E_1\rangle$ (exact physical ratio of pointer-state to eigenstate decay rate). In the thermodynamic limit the secular approximation fails, the doublet degenerates, and both rates converge. The three-regime structure is demonstrated in the Lipkin-Meshkov-Glick model via exact diagonalisation, and the algebraic origin of the discrepancy is established via the $\mathbb{Z}_2$ parity of the Lindblad jump operator.

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.

Desk Editor's Note private letter to a colleague

Parity symmetry removes the cross-term for energy eigenstates in these collective spin systems, producing protection factors of about 2.42 and 1.86.

read the letter

The one thing to take away is that parity symmetry explains the difference in Lindblad dephasing rates between localized pointer states and energy eigenstates in these collective spin systems. It sets the diagonal Jz matrix elements to zero for the eigenstates, removing the cross-term that doubles the rate for the pointers, and this leads to the two protection factors they define. What is new here is the explicit algebraic derivation from the parity of the jump operator and the definitions of eta_MF as (N m*)^2 over 2 G01 and eta_exact as (G01 + J01^2)/G01. That gives a clear handle on the basis dependence without fitting parameters. The paper does a solid job verifying this in the LMG model using exact diagonalization, mapping out the behavior near the critical crossover, in the ordered phase, and in the thermodynamic limit where both rates approach each other as the doublet becomes degenerate. The soft spots are limited. The argument assumes the Lindblad operator respects the Z2 parity and that the system is in the ordered phase with distinct bases; those are stated clearly but do restrict the scope. The numerical factors come straight from the model, and while the abstract reports 2.42 and 1.86, the full paper would need to show the convergence and any error estimates to make the numerics fully convincing. No internal issues with the algebra though. This is useful for people studying symmetry effects in open quantum many-body systems or collective spin decoherence. Readers working on quantum information with symmetric models or condensed matter open systems will get a practical distinction here. It has enough of a new result and a check to deserve serious referee attention. I would recommend sending it out for peer review. The derivation is direct and the model check supports it without overclaiming.

Referee Report

2 major / 2 minor

Summary. The paper claims that in the ordered phase of Z2-symmetric collective spin systems, Lindblad dephasing rates differ between localized pointer states {|P>, |R>} and energy eigenstates {|E0>, |E1>} by a factor approaching 2 as N→∞ (reaching ~2.42 near the quantum-critical crossover). The discrepancy originates algebraically from Z2 parity of the jump operator, which forces ⟨Ei|Jz|Ei⟩=0 exactly and eliminates the cross-term present for pointer states where ⟨P|Jz|P⟩=−⟨R|Jz|R⟩≠0. Two protection factors are derived: η_MF=(N m*)²/(2 G01)≈2.42 (mean-field advantage) and η_exact=(G01 + J01²)/G01≈1.86 (exact ratio), with G01 and J01 defined from the relevant matrix elements. The three-regime structure (including failure of the secular approximation in the thermodynamic limit) is demonstrated via exact diagonalization in the Lipkin-Meshkov-Glick model.

Significance. If the central algebraic identity holds, the work provides a parameter-free explanation of basis-dependent decoherence rates in symmetry-broken collective spins, with explicit strengths in the direct derivation from parity and matrix elements of the jump operator together with reproducible exact-diagonalization checks. It clarifies how symmetry protects certain coherences and identifies the breakdown of standard approximations at large N, offering falsifiable predictions for the quoted numerical factors in the LMG model.

major comments (2)

[Algebraic derivation (near Eqs. for G01 and J01)] The algebraic origin of the discrepancy (parity forcing ⟨Ei|Jz|Ei⟩=0 and removal of the cross-term) is load-bearing for the central claim; the manuscript should expand the explicit steps deriving the dephasing-rate expressions before and after applying this identity, including the precise definition of the rate formula used to obtain η_MF and η_exact.
[LMG-model numerical results section] The exact-diagonalization verification of the factors 2.42 and 1.86 near the crossover is only partially supported without accompanying data tables, error estimates, or finite-size scaling details; these are needed to confirm the numerical values are not sensitive to truncation or basis choice.

minor comments (2)

Notation for the order parameter m* and the matrix elements G01, J01 should be defined explicitly at first use in the main text rather than relying solely on the abstract.
[Thermodynamic-limit analysis] The discussion of secular-approximation failure in the thermodynamic limit would benefit from a brief statement of the precise validity condition (e.g., relative to the doublet splitting) with a supporting equation reference.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive assessment and constructive comments, which we address point by point below. We agree that both suggested expansions will improve clarity and reproducibility, and we will incorporate them in the revised manuscript.

read point-by-point responses

Referee: [Algebraic derivation (near Eqs. for G01 and J01)] The algebraic origin of the discrepancy (parity forcing ⟨Ei|Jz|Ei⟩=0 and removal of the cross-term) is load-bearing for the central claim; the manuscript should expand the explicit steps deriving the dephasing-rate expressions before and after applying this identity, including the precise definition of the rate formula used to obtain η_MF and η_exact.

Authors: We agree that expanding the derivation will strengthen the central claim. In the revised manuscript we will insert a dedicated paragraph (or short subsection) immediately after the definitions of G01 and J01. This addition will start from the Lindblad master equation in the interaction picture, apply the secular approximation to obtain the dephasing rates, explicitly compute the rates for both the pointer-state and energy-eigenstate bases, and then apply the parity identity ⟨Ei|Jz|Ei⟩=0 step by step to show the elimination of the cross term. We will also state the exact rate formula from which η_MF and η_exact are obtained. revision: yes
Referee: [LMG-model numerical results section] The exact-diagonalization verification of the factors 2.42 and 1.86 near the crossover is only partially supported without accompanying data tables, error estimates, or finite-size scaling details; these are needed to confirm the numerical values are not sensitive to truncation or basis choice.

Authors: We agree that additional numerical documentation is warranted for full reproducibility. In the revised manuscript we will add a table (or supplementary table) reporting the computed dephasing rates and protection factors for multiple system sizes N near the crossover, include error estimates arising from the diagonalization routine, and provide a short finite-size scaling discussion demonstrating convergence of the quoted factors. We will also explicitly state the basis truncation employed and verify that it is sufficient for the reported precision. revision: yes

Circularity Check

0 steps flagged

Derivation is algebraically self-contained with no circularity

full rationale

The central claims follow from the Lindblad master equation applied to a Z2-symmetric collective spin Hamiltonian. The key identity—that parity of the jump operator forces ⟨Ei|Jz|Ei⟩=0 exactly for energy eigenstates, removing the cross-term that appears for localized pointer states—is an immediate algebraic consequence of the stated symmetry assumption and does not rely on any fitted parameter or prior result. The protection factors are introduced by explicit definition: η_MF is written in terms of the order parameter m* and the averaged variance G01, while η_exact is written in terms of G01 and the off-diagonal matrix element J01; both quantities are computed directly from the model’s eigenstates. Numerical values (≈2.42, ≈1.86) and the thermodynamic-limit convergence are obtained as outputs of exact diagonalization in the LMG model, serving as verification rather than inputs. No self-citation chain, ansatz smuggling, or renaming of known results occurs; the three-regime structure is derived from the same parity and degeneracy properties. The derivation therefore remains independent of its own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the assumption that the decoherence is captured by a parity-respecting Lindblad operator and that the system remains in the symmetry-broken ordered phase; no free parameters are introduced beyond the model definition itself.

axioms (2)

domain assumption The environmental coupling is described by a Lindblad master equation whose jump operator commutes with the Z2 parity operator.
Invoked explicitly as the algebraic origin of the rate discrepancy via the Z2 parity of the Lindblad jump operator.
domain assumption The system is in the ordered phase where the two bases (pointer states and energy eigenstates) remain distinct.
Required for the reported factor-of-two difference and the protection factors to be meaningful.

pith-pipeline@v0.9.0 · 5613 in / 1581 out tokens · 43248 ms · 2026-05-09T20:01:00.672262+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

16 extracted references · 16 canonical work pages · 1 internal anchor

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S.Mouslopoulos,Classically Forbidden Signatures of Quantum Coherence in the Meso- scopic Lipkin–Meshkov–Glick Model, arXiv:2604.18638[quant-ph] (2026)

work page internal anchor Pith review Pith/arXiv arXiv 2026

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W. H. Zurek,Decoherence, einselection, and the quantum origins of the classical, Rev. Mod. Phys.75, 715 (2003)

work page 2003

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D. A. Lidar, I. L. Chuang, and K. B. Whaley,Decoherence-free subspaces for quantum computation, Phys. Rev. Lett.81, 2594 (1998)

work page 1998

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Morrison and A

S. Morrison and A. S. Parkins,Collective spin systems in dispersive optical cavity QED: Quantum phase transitions and entanglement, Phys. Rev. A77, 043810 (2008)

work page 2008

[5] [5]

Trenkwalder, G

A. Trenkwalder, G. Spagnolli, G. Semeghini, S. Coop, M. Landini, P. Castilho, L. Pezzè, G. Modugno, M. Inguscio, A. Smerzi, and M. Fattori,Quantum phase transi- tions with parity-symmetry breaking and hysteresis, Nat. Phys.12, 826 (2016)

work page 2016

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Botet, R

R. Botet, R. Jullien, and P. Pfeuty,Size scaling for infinitely coordinated systems, Phys. Rev. Lett.49, 478 (1982)

work page 1982

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Relaño, J

A. Relaño, J. M. Arias, J. Dukelsky, J. E. García-Ramos, and P. Pérez-Fernández, Decoherence as a signature of an excited-state quantum phase transition, Phys. Rev. A78, 060102(R) (2008)

work page 2008

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Tegmark,Importance of quantum decoherence in brain processes, Phys

M. Tegmark,Importance of quantum decoherence in brain processes, Phys. Rev. E61, 4194 (2000)

work page 2000

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D. J. Wineland, J. J. Bollinger, W. M. Itano, F. L. Moore, and D. J. Heinzen,Spin squeezing and reduced quantum noise in spectroscopy, Phys. Rev. A46, R6797 (1992)

work page 1992

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Kitagawa and M

M. Kitagawa and M. Ueda,Squeezed spin states, Phys. Rev. A47, 5138 (1993)

work page 1993

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L.Pezzè, A.Smerzi, M.K.Oberthaler, R.Schmied, andP.Treutlein,Quantum metrol- ogy with nonclassical states of atomic ensembles, Rev. Mod. Phys.90, 035005 (2018)

work page 2018

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A. J. Leggett and A. Garg,Quantum mechanics versus macroscopic realism: Is the flux there when nobody looks?, Phys. Rev. Lett.54, 857 (1985)

work page 1985

[13] [13]

Emary, N

C. Emary, N. Lambert, and F. Nori,Leggett–Garg inequalities, Rep. Prog. Phys.77, 016001 (2014)

work page 2014

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M. Kac, G. E. Uhlenbeck, and P. C. Hemmer,On the van der Waals Theory of the Vapor-Liquid Equilibrium. I. Discussion of a One-Dimensional Model, J. Math. Phys. 4, 216 (1963)

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Breuer and F

H.-P. Breuer and F. Petruccione,The Theory of Open Quantum Systems, Oxford University Press, Oxford (2002)

work page 2002

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H. J. Lipkin, N. Meshkov, and A. J. Glick,Validity of many-body approximation methods for a solvable model: (I). Exact solutions and perturbation theory, Nucl. Phys. 62, 188 (1965). 26

work page 1965