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arxiv: 2501.19269 · v2 · pith:JWE4I7T2new · submitted 2025-01-31 · 🌀 gr-qc · hep-th· quant-ph

Information Metrics and Possible Limitations of Local Information Objectivity in Quantum Gravity

Per Berglund , Andrew Geraci , Tristan Hubsch , David Mattingly , Djordje Minic This is my paper

Pith reviewed 2026-05-23 04:30 UTC · model grok-4.3

classification 🌀 gr-qc hep-thquant-ph
keywords quantum gravityinformation geometryFisher metricBorn ruleCencov's theoremgeneral covariancelocal information objectivity
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The pith

Deviations from the Fisher metric can modify the Born rule in quantum gravity, making it vary between observers.

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

The paper seeks to show that quantum gravity violates the conditions of Cencov's theorem, which uniquely determines the Fisher information metric on probability spaces under independent sampling. This violation permits other metrics that depend on the specific dynamical and environmental context of experiments. Because the form of the Born rule is heavily constrained by compatibility with such a metric, deviations can lead to an observer-dependent Born rule. This would extend the principle of general covariance from spacetime to information geometry. A reader should care because it questions whether fundamental quantum rules are the same for all local observers in a quantum gravitational setting.

Core claim

Local information objectivity relies on Cencov's theorem establishing the Fisher metric as unique. Quantum gravity typically violates the independent identically distributed sampling and sufficient statistics assumptions, allowing contextual deviations from the Fisher metric. These deviations can induce modifications of the Born rule, leading it to vary between observers, and suggest a new approach to quantum gravity via generally covariant information geometry.

What carries the argument

Deviations from the Fisher information metric on probability spaces, enabled by the failure of Cencov's theorem assumptions in quantum gravity contexts.

If this is right

  • Compatibility with non-Fisher metrics restricts the Born rule differently for different observers.
  • This provides an extension of spacetime general covariance to information geometry.
  • Experimental tests of possible variations in the Born rule are advocated.
  • A new quantum gravity approach based on generally covariant information geometry is suggested.

Where Pith is reading between the lines

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

  • If such metric deviations occur, they might affect how information is processed in strong gravitational fields.
  • This could link to questions about the consistency of quantum mechanics in curved spacetime.
  • Testable extensions might involve looking for observer-dependent probabilities in analog gravity systems.

Load-bearing premise

The dynamical and contextual features of quantum gravity violate the independent, identically distributed sampling and sufficient-statistics conditions of Cencov's theorem in a manner that permits non-Fisher metrics on probability space.

What would settle it

Observation that the Born rule remains invariant and identical for independent observers exchanging data in a quantum gravitational environment would falsify the proposed modifications.

read the original abstract

Local information objectivity, that local, independent observers can infer the same information about a model upon exchange of independently acquired experimental data, is fundamental to science. It is mathematically encoded via Cencov's theorem: the Fisher information metric is the unique metric invariant under the assumptions of independent, identically distributed sampling and sufficient statistics. However, quantum gravity typically violates these assumptions, permitting contextual deviations from the Fisher metric that reflect the dynamical experimental and environmental configurations. This yields a possible extension of spacetime general covariance to information geometry. Since compatibility with the metric on probability spaces heavily restricts the form of the Born rule for quantum mechanics, deviations from the Fisher metric also can induce modifications of the Born rule, leading it to vary between observers. We explain some possible variations, advocate for experimental tests, and suggest a new quantum gravity approach based on generally covariant information geometry.

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 / 0 minor

Summary. The manuscript claims that quantum gravity violates the i.i.d. sampling and sufficient-statistics assumptions of Cencov's theorem, permitting contextual deviations from the Fisher metric on probability space. These deviations are asserted to heavily restrict the form of the Born rule, rendering it observer-dependent and thereby extending spacetime general covariance to information geometry. The paper discusses possible variations of the rule, advocates experimental tests, and proposes a new quantum-gravity approach based on generally covariant information geometry.

Significance. If a rigorous, explicit mapping from non-Fisher metrics to modified Born-rule forms were supplied and shown to yield falsifiable predictions, the work would constitute a conceptually novel proposal at the quantum-gravity/information-geometry interface. At present the argument remains speculative and rests on an unshown logical step, so its significance is limited to raising a question rather than establishing a result.

major comments (2)
  1. [Abstract] Abstract (final paragraph): the central assertion that 'compatibility with the metric on probability spaces heavily restricts the form of the Born rule' and that 'deviations from the Fisher metric also can induce modifications of the Born rule' is load-bearing yet unsupported. No derivation, functional equation, or explicit replacement rule is given showing how a different metric on the probability simplex forces a different map p_i(ψ) from state vectors to probabilities. The standard construction proceeds in the opposite direction (Born rule first, Fisher metric induced), and the paper supplies no independent argument reversing this order.
  2. [Abstract] Abstract (paragraph 3): the claim that quantum gravity 'typically violates' the i.i.d. and sufficient-statistics conditions of Cencov's theorem in a manner that permits non-Fisher metrics is asserted without an independent derivation or external benchmark demonstrating that the resulting metric deviation actually alters the Born rule rather than merely reparameterizing the same probabilities.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for their careful reading and for identifying the key points where the manuscript's conceptual claims require clarification. The paper is exploratory in nature and does not claim to supply a complete derivation; we address the major comments below and indicate where revisions will be made.

read point-by-point responses

  1. Referee: [Abstract] Abstract (final paragraph): the central assertion that 'compatibility with the metric on probability spaces heavily restricts the form of the Born rule' and that 'deviations from the Fisher metric also can induce modifications of the Born rule' is load-bearing yet unsupported. No derivation, functional equation, or explicit replacement rule is given showing how a different metric on the probability simplex forces a different map p_i(ψ) from state vectors to probabilities. The standard construction proceeds in the opposite direction (Born rule first, Fisher metric induced), and the paper supplies no independent argument reversing this order.

    Authors: We agree that the manuscript does not contain an explicit functional equation or derivation that reverses the usual order from Born rule to metric. The central claim is presented as a logical consequence of Cencov's theorem combined with the expectation that quantum gravity breaks its assumptions, rather than as a proven result. We will revise the abstract to replace 'heavily restricts' with 'suggests possible restrictions' and add a short discussion paragraph noting the direction of the implication and the need for future explicit constructions. revision: yes

  2. Referee: [Abstract] Abstract (paragraph 3): the claim that quantum gravity 'typically violates' the i.i.d. and sufficient-statistics conditions of Cencov's theorem in a manner that permits non-Fisher metrics is asserted without an independent derivation or external benchmark demonstrating that the resulting metric deviation actually alters the Born rule rather than merely reparameterizing the same probabilities.

    Authors: The full text motivates the violation through the observer-dependent and background-independent character of measurements in quantum gravity, which precludes a fixed i.i.d. structure across observers. We acknowledge, however, that the manuscript does not supply a concrete model or benchmark separating genuine alteration of the Born rule from reparameterization. We will revise the relevant paragraph to state the claim as a hypothesis and include a brief remark distinguishing the two possibilities. revision: partial

standing simulated objections not resolved
  • An explicit, rigorous mapping from a general non-Fisher metric on the probability simplex to a modified Born rule p_i(ψ) that yields falsifiable predictions.

Circularity Check

0 steps flagged

No significant circularity detected; derivation remains conceptual and externally anchored.

full rationale

The paper anchors its starting point in Cencov's theorem, an established external mathematical result on uniqueness of the Fisher metric under i.i.d. and sufficient-statistic conditions. It then hypothesizes that quantum gravity may violate those conditions, permitting non-Fisher metrics, and notes that metric compatibility restricts the Born rule. No equations, fitted parameters, or self-citations are exhibited that reduce the claimed observer-dependence of the Born rule to the violation assumption by construction. The argument is presented as a possible implication open to experimental tests rather than a closed derivation whose output is definitionally identical to its input. This is the normal case of a self-contained conceptual proposal.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the premise that quantum gravity necessarily violates the IID and sufficient-statistics conditions of Cencov's theorem in a way that permits non-unique metrics; no free parameters or new entities are introduced in the abstract, but the violation itself functions as an unproven domain assumption.

axioms (1)
  • domain assumption Quantum gravity violates the independent, identically distributed sampling and sufficient-statistics assumptions underlying Cencov's theorem.
    Invoked in abstract paragraph 3 as the reason deviations from the Fisher metric are permitted.

pith-pipeline@v0.9.0 · 5683 in / 1266 out tokens · 26748 ms · 2026-05-23T04:30:05.160696+00:00 · methodology

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Quantum Spacetime, Quantum Gravity and Gravitized Quantum Theory

    gr-qc 2026-04 unverdicted novelty 5.0

    Quantum spacetime with a non-commutative dual explains the fixed Born rule of quantum theory and leads to gravitized quantum mechanics featuring dynamical probabilities and higher-order interference.

Reference graph

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