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arxiv: 2312.14516 · v2 · submitted 2023-12-22 · 🌀 gr-qc · hep-th

Non-classicality of Primordial Gravitational Waves in Three-mode Representation Through Quantum Poincare Sphere

Anom Trenggana , Freddy P. Zen This is my paper

Pith reviewed 2026-05-24 05:06 UTC · model grok-4.3

classification 🌀 gr-qc hep-th
keywords primordial gravitational wavesthree-mode Bogoliubov transformationquantum discordquantum Poincare spherenon-classicalitysqueezed statesearly universe
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The pith

Generalizing to three-mode Bogoliubov transformations lets primordial gravitational waves become classical at large squeezing if only two modes are considered.

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

This paper generalizes the standard two-mode Bogoliubov transformation for creating primordial gravitational waves from the vacuum to a three-mode version. Using quantum discord as a measure, it finds that the universe can behave classically for large squeezing parameters if only two of the three modes are taken into account. The quantum Poincare sphere, used to track quantum features, gives non-classical results whenever the squeezing parameter is positive, matching the two-mode case. When the initial state is a coherent state, the sphere no longer depends on the squeezing parameter and instead shows non-classicality whenever cosθ or sinθ is nonzero.

Core claim

We generalize the vacuum state transformation that generates gravitational waves to a three-mode Bogoliubov transformation. Quantum discord calculations reveal that classicality emerges at large squeezed parameters when restricting to two modes. The quantum Poincare sphere indicates non-classical characteristics if the squeezed parameter exceeds zero, consistent with two-mode results. For coherent initial states, the sphere becomes independent of the squeezed parameter and non-classical provided cosθ or sinθ is not zero.

What carries the argument

Three-mode Bogoliubov transformation, diagnosed via quantum discord for classicality and the quantum Poincare sphere for quantum characteristics.

If this is right

  • The universe appears classical for sufficiently large squeezing under the two-mode restriction of the three-mode model.
  • Non-classical features are detected by the quantum Poincare sphere for any positive squeezing parameter in the general case.
  • Coherent initial states lead to non-classicality that depends on the angles θ rather than the squeezing parameter.
  • The quantum characteristics of gravitational waves remain similar to the two-mode case when using the three-mode representation.

Where Pith is reading between the lines

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

  • Multi-mode effects in this framework might help reconcile the quantum origin of fluctuations with the classical appearance of the late universe.
  • Measurements of gravitational wave correlations could test whether angle-dependent non-classicality appears in coherent-state scenarios.
  • Considering additional modes beyond three could further map out the parameter regions where classicality sets in.

Load-bearing premise

The three-mode Bogoliubov transformation accurately models the generation of gravitational waves from the vacuum in the early universe, and quantum discord plus the Poincare sphere reliably distinguish classical from non-classical states.

What would settle it

A calculation or observation showing quantum discord remains nonzero at large squeezing even when restricting the three-mode model to two modes, or that the Poincare sphere depends on squeezing for coherent initial states.

Figures

Figures reproduced from arXiv: 2312.14516 by Anom Trenggana, Freddy P. Zen.

Figure 1
Figure 1. Figure 1: The plot of the quantum discord equation (4.8) against the squeezed parameter (rk). time (η). This means that in the radiation domination era, the squeezed parameter will have a very high value and it can be said also that the correlation in the two-mode representation is highly quantum in nature. These results raise a question about how the universe could become classical in the era of radiation dominatio… view at source ↗
Figure 2
Figure 2. Figure 2: The plot of the tripartite quantum discord equation (4.13) toward the squeezed parameter (rk) with varying parameters θ. Where for θ = 0.5π the value of the quantum discord will be the same as in the case of two-mode representation. of the two-mode representation. From these results, it can also be seen that the correlation of the three modes will be quantum if the squeezed parameter has a value of r > 0. … view at source ↗
Figure 3
Figure 3. Figure 3: The plot of quantum discord when we observe only two of the three modes toward the parameter θ with the squeezed parameter r = 1. Then the results of these quantum discord measurements will be plotted toward the parameter θ where the squeezed parameter r = 1 as the figure (3). It can be seen that the mode (2 & 3) has results that are independent of the parameter θ. Where the quantum discord will have the s… view at source ↗
Figure 4
Figure 4. Figure 4: Poincare sphere illustration based on equation (5.2). [PITH_FULL_IMAGE:figures/full_fig_p015_4.png] view at source ↗
read the original abstract

In this research, we generalize the transformation of the vacuum state that generated gravitational waves in the early universe which is usually transformed using a two-mode into a three-mode Bogoliubov transformation. Based on the calculation of quantum discord this transformation allows the universe to be classical when the squeezed parameter is large if only of the three possible modes, only two are considered. We also studied the quantum characteristics of those gravitational waves by calculating an observable quantity named the quantum Poincare sphere. The result will be the same as the two-mode transformation, where quantum characteristics appear if the squeezed parameter is greater than zero. However, if the initial state is coherent, different results will be obtained, the quantum Poincare sphere will not depend on the squeezed parameter and will be non-classical if $\cos\theta$ or $\sin\theta$ is not zero.

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

3 major / 3 minor

Summary. The manuscript generalizes the standard two-mode Bogoliubov transformation for primordial gravitational waves to a three-mode version. Based on quantum discord calculations, it claims that the system can appear classical for large squeezing parameter r when only two of the three modes are retained. It further computes the quantum Poincaré sphere, reporting that non-classicality appears for squeezed states whenever r > 0 (matching the two-mode case), while for coherent initial states the sphere is independent of r and non-classical provided cos θ or sin θ is nonzero.

Significance. If the three-mode transformation were shown to arise from the tensor perturbation equations, the discord-based classicality result and the Poincaré-sphere diagnostics could extend existing discussions of the quantum-to-classical transition for primordial GWs. The work supplies explicit parameter dependence (r, θ) that could be tested against future observations, but the absence of a cosmological derivation limits its immediate applicability.

major comments (3)
  1. [Introduction and §2] Introduction and §2 (Bogoliubov transformation): the three-mode generalization is stated without derivation from the quadratic action for tensor perturbations in FLRW or from the resulting mode equations. Standard cosmology employs two-mode squeezing for the two polarization degrees of freedom; without this link the subsequent claim that restricting to two modes yields classicality at large r lacks a physical basis.
  2. [Quantum-discord section (likely §3)] Quantum-discord section (likely §3): the assertion that the universe becomes classical for large r when only two modes are kept rests on the discord vanishing. No explicit expression for the reduced density matrix or the discord formula is supplied in the abstract, and the full text must demonstrate that the reduction is not tautological.
  3. [Quantum Poincaré sphere section (likely §4)] Quantum Poincaré sphere section (likely §4): the statements that the sphere is independent of r for coherent states and non-classical when cos θ or sin θ ≠ 0 require the explicit operator or Stokes-parameter expressions used to define the sphere; without them it is impossible to verify consistency with the two-mode limit or to assess whether the result follows from the three-mode transformation.
minor comments (3)
  1. [Abstract] Abstract: the sentence 'if only of the three possible modes, only two are considered' is grammatically unclear and should be rephrased.
  2. [Abstract] Abstract: the phrase 'the result will be the same as the two-mode transformation' is vague; specify which observable and which limit is intended.
  3. [References] References: standard two-mode treatments of PGW squeezing (e.g., papers deriving the Bogoliubov coefficients from the Mukhanov-Sasaki equation) are not cited, making the generalization harder to contextualize.

Simulated Author's Rebuttal

3 responses · 1 unresolved

We thank the referee for the thorough review and constructive suggestions. We address each major comment below, indicating planned revisions where appropriate. Our responses focus on clarifying the manuscript's scope as a mathematical generalization while strengthening the presentation of calculations.

read point-by-point responses

  1. Referee: [Introduction and §2] Introduction and §2 (Bogoliubov transformation): the three-mode generalization is stated without derivation from the quadratic action for tensor perturbations in FLRW or from the resulting mode equations. Standard cosmology employs two-mode squeezing for the two polarization degrees of freedom; without this link the subsequent claim that restricting to two modes yields classicality at large r lacks a physical basis.

    Authors: We agree that the three-mode Bogoliubov transformation is presented as a direct generalization of the standard two-mode case without an explicit derivation from the quadratic action or mode equations in FLRW. The manuscript explores the quantum-information consequences of this extension, motivated by the two-mode squeezing that arises in standard treatments of tensor perturbations. We will revise the introduction and §2 to explicitly state that the three-mode form is a mathematical generalization chosen to investigate multi-mode effects, and we will add a brief discussion of how it reduces to the two-mode limit. A full derivation from the perturbation equations would require additional model assumptions not present in the current work. revision: partial

  2. Referee: [Quantum-discord section (likely §3)] Quantum-discord section (likely §3): the assertion that the universe becomes classical for large r when only two modes are kept rests on the discord vanishing. No explicit expression for the reduced density matrix or the discord formula is supplied in the abstract, and the full text must demonstrate that the reduction is not tautological.

    Authors: We will add the explicit form of the three-mode state after the Bogoliubov transformation, the reduced density matrix obtained by tracing over the third mode, and the full expression for quantum discord (including the von Neumann entropies involved) in the revised §3. This will allow readers to verify that the vanishing of discord at large r follows from the calculation rather than from the tracing procedure itself. revision: yes

  3. Referee: [Quantum Poincaré sphere section (likely §4)] Quantum Poincaré sphere section (likely §4): the statements that the sphere is independent of r for coherent states and non-classical when cos θ or sin θ ≠ 0 require the explicit operator or Stokes-parameter expressions used to define the sphere; without them it is impossible to verify consistency with the two-mode limit or to assess whether the result follows from the three-mode transformation.

    Authors: We will include the explicit definitions of the Stokes operators and the corresponding parameters used to construct the quantum Poincaré sphere in the revised §4. These expressions will be shown to recover the known two-mode results in the appropriate limit, and we will demonstrate how the three-mode transformation enters the calculation of the sphere for both squeezed and coherent initial states. revision: yes

standing simulated objections not resolved
  • The request for a derivation of the three-mode transformation directly from the quadratic action for tensor perturbations in FLRW, as the manuscript is an exploratory generalization of the Bogoliubov transformation rather than a first-principles cosmological derivation.

Circularity Check

0 steps flagged

No circularity: derivation uses independent diagnostics on an adopted generalization

full rationale

The paper adopts a three-mode Bogoliubov transformation as a generalization of the standard two-mode case, then computes quantum discord and the quantum Poincaré sphere as separate observables on the resulting states. No equations or steps reduce the claimed classicality condition (large squeeze parameter with two modes) or the sphere results (independence of r for coherent states when cosθ or sinθ ≠ 0) back to the inputs by definition or by renaming a fitted quantity. No self-citations are invoked as load-bearing uniqueness theorems, and the diagnostics are applied rather than presupposed. The chain therefore remains self-contained against external benchmarks for classicality measures.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The paper rests on standard assumptions from quantum field theory in curved spacetime and quantum optics applied to cosmology. No new entities are postulated. The squeeze parameter and angle θ are standard free parameters varied across regimes rather than fitted to new data.

free parameters (2)
  • squeezed parameter
    Standard parameter controlling squeezing strength in the Bogoliubov transformation; varied to explore classicality regimes.
  • θ
    Angle parameter appearing in the three-mode transformation and coherent-state results.
axioms (2)
  • domain assumption Bogoliubov transformations can be generalized from two to three modes while preserving the necessary commutation relations for describing vacuum fluctuations in the early universe.
    Invoked as the basis for extending the usual treatment of primordial gravitational waves.
  • domain assumption Quantum discord serves as a reliable indicator of whether a state (and thus the universe) is classical.
    Used to conclude classical behavior for large squeeze parameter when only two modes are considered.

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