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arxiv: 2605.13494 · v1 · submitted 2026-05-13 · 🪐 quant-ph

Recognition: 2 theorem links

· Lean Theorem

Violations of the Leggett-Garg inequality in Hybrid Liouvillian Dynamics: The Nonlinear Role of Detector Efficiency

Sourav Paul , Parveen Kumar , Sourin Das
Authors on Pith no claims yet

Pith reviewed 2026-05-14 17:55 UTC · model grok-4.3

classification 🪐 quant-ph
keywords Leggett-Garg inequalitydetector efficiencynon-Hermitian dynamicsopen quantum systemsquantum trajectorieshybrid LiouvillianLeggett-Garg correlator
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The pith

Even an infinitesimal detector efficiency causes a rapid nonlinear suppression of Leggett-Garg inequality violations from their algebraic maximum of 3 to the classical bound.

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

This paper investigates the robustness of Leggett-Garg inequality violations in an open two-level quantum system when realistic measurement efficiencies are included. It introduces a hybrid Liouvillian dynamics controlled by a continuous parameter q representing the fraction of quantum jumps retained by the detector. At q equal to zero, corresponding to ideal non-Hermitian no-jump evolution, the violation measure K3 can reach its algebraic maximum of three. However, any positive value of q, no matter how small, leads to a sharp drop in K3 toward the classical limit of one due to a highly nonlinear dependence. This result matters because it shows that extreme quantum violations in continuous dynamics are not stable but depend on nearly perfect post-selection of trajectories, imposing strict requirements on experimental detectors.

Core claim

The central claim is that in the hybrid Liouvillian framework, the Leggett-Garg parameter K3 reaches its algebraic bound of 3 only in the limit of zero detector efficiency. For any finite efficiency, the dynamics incorporate enough jump trajectories to suppress the violation in a logarithmic manner back to the classical bound of 1. This fragility arises within continuous, divisible quantum trajectory evolution and contrasts with more robust violations possible in discrete protocols.

What carries the argument

The hybrid Liouvillian superoperator parameterized by detector efficiency q, which blends Lindblad jump terms with non-Hermitian evolution to model partial retention of quantum trajectories.

If this is right

  • Extreme LGI violations demand near-perfect suppression of detected quantum jumps via effective post-selection.
  • Detector performance must be extremely high to access the algebraic bound in continuous evolution setups.
  • The nonlinear sensitivity implies that small improvements in efficiency yield large reductions in observed violations.
  • Such maximal violations are singular limits rather than generic features of the open-system dynamics.

Where Pith is reading between the lines

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

  • Similar nonlinear fragility might affect other temporal Bell inequalities under hybrid dynamics.
  • Experiments could probe this by tuning effective efficiency through post-selection strength and tracking K3.
  • This highlights the need for protocols that achieve strong violations without relying on idealized no-jump conditions.

Load-bearing premise

The chosen hybrid Liouvillian accurately captures how real detectors with finite efficiency affect the ensemble of quantum trajectories.

What would settle it

Perform an experiment on a two-level system with tunable detector efficiency and measure if K3 falls from near 3 to near 1 as efficiency increases from zero in the specific nonlinear way predicted by the model.

Figures

Figures reproduced from arXiv: 2605.13494 by Parveen Kumar, Sourav Paul, Sourin Das.

Figure 1
Figure 1. Figure 1: FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: , state trajectories collapse toward the mixed-state origin of the Bloch sphere. In this regime, K3 strictly obeys the L¨uders bound (K3 ≤ 1.5) and trends toward the classical limit (K3 → 1) as γ increases. Conversely, in the Non-Hermitian Limit (q ≪ 1), conditioning the ensemble on jump-free trajectories ro￾bustly stabilizes coherent evolution despite environmen￾tal coupling. The highly nonlinear renormal… view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
read the original abstract

Violations of the Leggett-Garg inequality (LGI) up to its algebraic bound under non-Hermitian dynamics are well established theoretically. Here, we demonstrate that such extreme violations are intrinsically fragile when realistic measurement processes are taken into account. We consider an open two-level system described by a time-local hybrid Liouvillian, with a continuous parameter $q \in [0,1]$, representing detector efficiency, i.e., the fraction of quantum jump trajectories that are retained in the ensemble. This parameter interpolates between trace-preserving Lindblad dynamics ($q=1$) and non-Hermitian ``no-jump" evolution ($q=0$). While $K_3$ approaches its algebraic maximum of 3 in the null-efficiency limit, even an infinitesimal increase in detector efficiency induces a rapid, highly nonlinear suppression toward the classical bound. This logarithmic sensitivity reveals that maximal LGI violations are not robust physical features but rather singular limits of idealized measurement conditions. Our results have direct experimental implications: achieving algebraic LGI violations in systems undergoing continuous time evolution requires near-perfect suppression of detected quantum jumps (i.e., effective post-selection), placing stringent constraints on detector performance. In contrast to discrete protocols based on time-non-divisible dynamics, our framework shows that extreme violations arising within continuous, divisible quantum trajectory evolution constitute a fundamentally fragile regime.

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

1 major / 1 minor

Summary. The manuscript claims that extreme violations of the Leggett-Garg inequality (K3 approaching its algebraic bound of 3) under non-Hermitian dynamics are fragile when modeled with a time-local hybrid Liouvillian incorporating a continuous detector-efficiency parameter q ∈ [0,1]. This parameter interpolates between full Lindblad dynamics (q=1) and no-jump non-Hermitian evolution (q=0), with the central result being a rapid, highly nonlinear suppression of K3 toward the classical bound even for infinitesimal q > 0.

Significance. If the hybrid Liouvillian construction is accepted as a faithful model of realistic continuous measurements, the result demonstrates that algebraic LGI violations constitute a singular limit requiring near-perfect post-selection, imposing stringent experimental constraints on detector performance in continuous-time protocols. The work supplies a concrete parameterization and highlights a logarithmic sensitivity that distinguishes continuous divisible dynamics from discrete non-Markovian protocols.

major comments (1)
  1. [§II] §II (Hybrid Liouvillian definition): The central claim that K3 exhibits rapid nonlinear suppression for any q>0 rests on a specific interpolation in which only the jump term is scaled by q while the anticommutator term is retained in full. This construction does not reproduce the unconditional master equation of standard quantum-trajectory theory for inefficient detection (efficiency η=q), where undetected jumps still contribute to the Lindblad form and the ensemble-averaged ρ(t) remains independent of η. The reported fragility may therefore be an artifact of the chosen parameterization rather than a generic feature of realistic detector inefficiency.
minor comments (1)
  1. [Abstract] The abstract refers to 'logarithmic sensitivity' without specifying whether this is obtained analytically from the characteristic equation or observed numerically; the main text should clarify the derivation of the functional form of K3(q) near q=0.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their insightful comments on our manuscript. We appreciate the opportunity to clarify the physical motivation behind our hybrid Liouvillian construction. We address the major comment in detail below.

read point-by-point responses

  1. Referee: [§II] §II (Hybrid Liouvillian definition): The central claim that K3 exhibits rapid nonlinear suppression for any q>0 rests on a specific interpolation in which only the jump term is scaled by q while the anticommutator term is retained in full. This construction does not reproduce the unconditional master equation of standard quantum-trajectory theory for inefficient detection (efficiency η=q), where undetected jumps still contribute to the Lindblad form and the ensemble-averaged ρ(t) remains independent of η. The reported fragility may therefore be an artifact of the chosen parameterization rather than a generic feature of realistic detector inefficiency.

    Authors: We thank the referee for this observation, which allows us to better articulate the scope of our model. Our hybrid Liouvillian is specifically designed to interpolate between the full Lindblad master equation (q = 1) and the non-Hermitian no-jump evolution (q = 0) by scaling only the jump operator term while retaining the full anticommutator. This choice is physically motivated by a continuous detector efficiency parameter q, defined as the fraction of quantum jump trajectories retained in the ensemble average. In this framework, q = 1 corresponds to including all trajectories (standard Lindblad), while q = 0 corresponds to post-selecting exclusively on no-jump trajectories, yielding the non-Hermitian dynamics. This is distinct from the unconditional master equation in standard quantum trajectory theory for inefficient detectors, where the ensemble average is indeed η-independent. However, our interest lies in the regime where partial retention of jumps (small but nonzero q) is considered, which models realistic continuous monitoring with imperfect post-selection. The rapid nonlinear suppression of K3 for q > 0 demonstrates the fragility of algebraic violations under such conditions, highlighting the need for near-perfect jump suppression. We have revised §II to include an explicit discussion of this distinction and the physical interpretation of q, ensuring the model is not misinterpreted as the unconditional case. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper introduces a hybrid Liouvillian with explicit continuous parameter q and derives the behavior of K3 by direct solution of the resulting master equation across q in [0,1]. The reported nonlinear suppression is obtained by evaluating the Leggett-Garg correlator on the interpolated dynamics; no parameter is fitted to data, no result is renamed as a prediction, and no load-bearing step reduces to a self-citation or self-definition. The derivation remains self-contained against the stated model equations.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on the standard definition of the Leggett-Garg correlator K3, the existence of a time-local hybrid Liouvillian that interpolates via q, and the assumption that q represents realistic partial detection of jumps.

free parameters (1)
  • q
    Continuous detector-efficiency parameter introduced to interpolate between Lindblad (q=1) and non-Hermitian (q=0) regimes.
axioms (1)
  • domain assumption The hybrid Liouvillian dynamics with parameter q faithfully models open two-level systems under partial quantum-jump detection.
    Invoked to define the family of evolutions studied.

pith-pipeline@v0.9.0 · 5545 in / 1162 out tokens · 51367 ms · 2026-05-14T17:55:35.844963+00:00 · methodology

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Reference graph

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