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arxiv: 2604.09772 · v1 · submitted 2026-04-10 · 🪐 quant-ph · cond-mat.stat-mech· hep-th

Leggett-Garg Inequality Violations Bound Quantum Fisher Information

Nick Abboud , Yuntao Guan , Barry Bradlyn , Jorge Noronha This is my paper

Pith reviewed 2026-05-10 18:07 UTC · model grok-4.3

classification 🪐 quant-ph cond-mat.stat-mechhep-th
keywords Leggett-Garg inequalityquantum Fisher informationquantum metrologyentanglement depthmany-body coherencetemporal correlationsstationary states
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The pith

Violations of Leggett-Garg inequalities bound the quantum Fisher information from below in stationary pure and thermal states.

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

The paper shows that when a Leggett-Garg inequality is violated by bounded observables in stationary pure states or thermal states, this violation directly implies a lower bound on the system's quantum Fisher information. This connection transforms a test for macroscopic realism into a tool for quantifying quantum metrological advantage. Sympathetic readers would care because it provides an experimentally accessible way to witness quantum coherence and entanglement depth using only temporal correlations of one observable, without needing complete state tomography. The work also ties these violations to the same spectral properties and sum rules that govern many-body response functions.

Core claim

We prove that a violation of a Leggett-Garg inequality for bounded observables in stationary pure states and thermal states yields a rigorous lower bound on the quantum Fisher information. This turns a qualitative foundations test of realism in quantum systems into a quantitative witness of useful quantum sensitivity and, in the collective setting, into a lower bound on multipartite entanglement depth in many-body systems. We further demonstrate that Leggett-Garg violations are constrained by the same spectral moments, susceptibilities, and f-sum-rule bounds that organize many-body response.

What carries the argument

The derived lower bound on quantum Fisher information expressed in terms of the magnitude of Leggett-Garg inequality violation for bounded observables.

Load-bearing premise

The derivation assumes stationary pure states and thermal states together with bounded observables; if these restrictions are lifted or if the states are non-stationary, the bound may not hold.

What would settle it

An explicit counterexample of a stationary pure state or thermal state in which a Leggett-Garg inequality is violated yet the quantum Fisher information lies below the claimed lower bound would falsify the central result.

Figures

Figures reproduced from arXiv: 2604.09772 by Barry Bradlyn, Jorge Noronha, Nick Abboud, Yuntao Guan.

Figure 1
Figure 1. Figure 1: FIG. 1. Plot of [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
read the original abstract

We prove that a violation of a Leggett-Garg inequality for bounded observables in stationary pure states and thermal states yields a rigorous lower bound on the quantum Fisher information. This turns a qualitative foundations test of realism in quantum systems into a quantitative witness of useful quantum sensitivity and, in the collective setting, into a lower bound on multipartite entanglement depth in many-body systems. We further demonstrate that Leggett-Garg violations are constrained by the same spectral moments, susceptibilities, and $f$-sum-rule bounds that organize many-body response. Our results show that temporal correlations of a single collective observable can serve as an experimentally accessible witness of many-body quantum coherence, without requiring full state reconstruction.

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

0 major / 3 minor

Summary. The manuscript proves that a violation of a Leggett-Garg inequality for bounded observables in stationary pure states and thermal states yields a rigorous lower bound on the quantum Fisher information. This converts the LGI test into a quantitative witness of quantum sensitivity and, for collective observables, a lower bound on multipartite entanglement depth. The work further shows that such violations are constrained by the same spectral moments, susceptibilities, and f-sum-rule bounds that govern many-body linear response.

Significance. If the central derivation holds, the result is significant because it supplies an experimentally accessible route from temporal correlation measurements to quantitative bounds on metrological usefulness and entanglement depth without requiring full tomography. The explicit linkage to response-theory quantities (moments, susceptibilities, f-sum rules) is a clear strength, as it embeds the bound in a well-established many-body framework and thereby increases its applicability to condensed-matter and quantum-optical platforms.

minor comments (3)
  1. Abstract: the phrase 'in the collective setting' is used without specifying the precise form of the collective observable or the scaling of the entanglement-depth bound; a single clarifying sentence would improve readability.
  2. The manuscript would benefit from an explicit statement (near the main theorem) of the numerical prefactor relating the LGI violation magnitude to the QFI lower bound, even if the derivation is parameter-free within the stated domain.
  3. Figure captions (if present) should indicate whether the plotted curves are for pure states, thermal states, or both, to allow immediate comparison with the analytic bounds.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the positive assessment of its significance. The recommendation for minor revision is noted, and we appreciate the recognition that the linkage to response theory and the experimental accessibility of the bounds are strengths. As the report lists no specific major comments, we have no point-by-point rebuttals to provide.

Circularity Check

0 steps flagged

No significant circularity; derivation is a direct proof under restricted assumptions

full rationale

The paper presents a mathematical proof that Leggett-Garg inequality violations for bounded observables imply a lower bound on quantum Fisher information, but only for stationary pure states and thermal states. The abstract and claim structure explicitly restrict the domain, and no equations or steps reduce by construction to fitted parameters, self-definitions, or load-bearing self-citations. The central result is a rigorous implication within the stated setting rather than a renaming or ansatz smuggling. This is the common case of a self-contained derivation against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard quantum mechanics for stationary and thermal states plus the boundedness of observables. No free parameters, new entities, or ad-hoc axioms are introduced in the abstract.

axioms (2)
  • domain assumption The system is described by standard quantum mechanics in stationary pure or thermal states
    Invoked to restrict the domain where the Leggett-Garg violation implies the Fisher-information bound.
  • domain assumption Observables under consideration are bounded
    Explicitly required for the inequality violation to produce the stated lower bound.

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Works this paper leans on

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