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arxiv: 2605.20995 · v1 · pith:7PVCC3TCnew · submitted 2026-05-20 · ✦ hep-th · gr-qc· quant-ph

Wasserstein Space of Quantum Chaos

Koji Hashimoto , Norihiro Tanahashi , Kentaroh Yoshida This is my paper

Pith reviewed 2026-05-21 04:00 UTC · model grok-4.3

classification ✦ hep-th gr-qcquant-ph
keywords Wasserstein distancequantum chaosoptimal transportHusimi Q-representationquantum scramblingLyapunov exponentquantum scarseffective dimension
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The pith

Effective dimension of the Wasserstein space of energy eigenstates decreases as quantum systems grow more chaotic.

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

The paper establishes that greater quantum chaos corresponds to a lower effective dimension in the space formed by Wasserstein distances between energy eigenstates. The authors reach this conclusion by representing states via Husimi Q-functions in a coupled harmonic oscillator model, applying Sinkhorn-regularized optimal transport to obtain distances, and then using Gram-spectrum embedding to extract the geometry. Exponential growth of out-of-time-order correlators creates folding in this space that may drive the dimension reduction. The same construction recovers the Lyapunov exponent at the separatrix of an inverted oscillator and identifies quantum scars through branching patterns. A reader would care because the method supplies a geometric diagnostic for scrambling, chaos, and scars while linking to ideas about emergent holographic spaces.

Core claim

The effective dimension of the Wasserstein space of energy eigenstates decreases as a quantum system becomes more chaotic. Exponential OTOC growth induces a folding structure in the emergent Wasserstein space. At the separatrix of the inverted harmonic oscillator the Wasserstein distance captures the Lyapunov exponent. Branching structures in the space signal quantum scar states.

What carries the argument

Sinkhorn-regularized optimal transport distances computed on Husimi Q-representations of energy eigenstates, then embedded via the Gram-spectrum method to produce the Wasserstein space geometry.

If this is right

  • Exponential OTOC growth produces folding that lowers the effective dimension of the Wasserstein space.
  • The Wasserstein distance measured at the separatrix of the inverted oscillator equals the classical Lyapunov exponent.
  • Branching patterns in the Wasserstein space mark the locations of quantum scar states inside the chaotic region of phase space.
  • The observed dimensional reduction supports the conjecture that the Wasserstein space functions as an emergent holographic geometry.

Where Pith is reading between the lines

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

  • The folding mechanism may generalize to other measures of scrambling and provide a geometric criterion for the onset of chaos in many-body systems.
  • Applying the same construction to lattice gauge theories or holographic models could test whether the dimension reduction tracks black-hole-like behavior.
  • The branching signature of scars might be used to locate scarred eigenstates without scanning the full spectrum.

Load-bearing premise

The Gram-spectrum embedding of these regularized transport distances on Husimi representations faithfully tracks the underlying chaotic dynamics without dominant artifacts from regularization strength or the choice of phase-space function.

What would settle it

Compute the effective dimension for a well-studied chaotic system such as the quantum stadium billiard using the same Husimi-plus-Sinkhorn procedure and find that the dimension does not drop relative to an integrable counterpart.

read the original abstract

We find that the effective dimension of the Wasserstein space of energy eigenstates decreases as a quantum system becomes more chaotic. To demonstrate this, we study a quantum coupled harmonic oscillator system using Husimi Q-representations, to which Sinkhorn-regularized optimal transport is applied to construct an embedding geometry via the Gram-spectrum method. We also demonstrate that exponential OTOC growth, referred to here as quantum scrambling even in the absence of chaos, induces a folding structure in the emergent Wasserstein space, which may underlie the chaotic reduction of the Wasserstein dimension. At the separatrix (the scrambling point) of the inverted harmonic oscillator, the Wasserstein distance correctly captures the Lyapunov exponent. Furthermore, we discover that a branching structure in the Wasserstein space signals quantum scar states within the chaotic sea of phase space. Our optimal transport approach thus provides a new diagnostic for quantum chaos, quantum scrambling, quantum scars, and quantum Lyapunov exponents. The observed chaotic dimensional reduction also supports the recent conjecture [arXiv:2604.17649] that the Wasserstein space serves as an emergent holographic space through the manifold hypothesis, since chaoticity is a characteristic signature of black holes in holography.

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 / 2 minor

Summary. The manuscript claims that the effective dimension of the Wasserstein space of energy eigenstates, constructed via Sinkhorn-regularized optimal transport on Husimi Q-representations, decreases as a quantum system becomes more chaotic. Using a coupled harmonic oscillator model, it reports that exponential OTOC growth induces folding in the emergent space, that the Wasserstein distance captures the Lyapunov exponent at the separatrix of the inverted oscillator, and that branching structures signal quantum scar states. These observations are presented as a new diagnostic for quantum chaos and as numerical support for the conjecture that Wasserstein space realizes an emergent holographic geometry via the manifold hypothesis.

Significance. If the numerical trends survive robustness checks, the work introduces an innovative application of optimal transport geometry to quantum phase-space distributions, yielding concrete diagnostics for chaos, scrambling, and scars. The reported dimensional reduction in chaotic regimes supplies a falsifiable numerical test of the manifold hypothesis for emergent holographic space, an idea that connects directly to black-hole physics in holography. The approach is technically novel in combining Sinkhorn regularization with Gram-spectrum embeddings and could stimulate further cross-fertilization between quantum information geometry and holographic methods.

major comments (2)
  1. [Abstract / Numerical methods] Abstract and numerical methods: the headline claim that effective dimension decreases with chaos is obtained from Gram-spectrum embeddings of Sinkhorn-regularized OT distances on Husimi Q-functions. No systematic scan over the regularization parameter or comparison with an alternative phase-space representation (e.g., Wigner) is reported. Because the entropic penalty can preferentially smooth distances among delocalized chaotic states, it is unclear whether the observed dimensional reduction is driven by the underlying dynamics or by the numerical scheme; this is load-bearing for the central result.
  2. [Abstract] Abstract: the assertion that the Wasserstein distance 'correctly captures the Lyapunov exponent' at the separatrix requires quantitative comparison (e.g., extracted exponent versus classical value, error bars, and dependence on state selection). Without these details the agreement remains qualitative and does not yet establish the method as a reliable quantum Lyapunov diagnostic.
minor comments (2)
  1. [Numerical Setup] The manuscript would benefit from explicit tabulation of all Hamiltonian parameters, basis sizes, and Sinkhorn convergence tolerances used in the coupled-oscillator calculations.
  2. [Figures] Figure captions should clearly label which curves or embeddings correspond to regular versus chaotic regimes and indicate the cutoff criterion employed for the effective dimension.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading, positive assessment of the work's novelty, and constructive suggestions. We address each major comment below and will incorporate revisions to strengthen the manuscript.

read point-by-point responses

  1. Referee: [Abstract / Numerical methods] Abstract and numerical methods: the headline claim that effective dimension decreases with chaos is obtained from Gram-spectrum embeddings of Sinkhorn-regularized OT distances on Husimi Q-functions. No systematic scan over the regularization parameter or comparison with an alternative phase-space representation (e.g., Wigner) is reported. Because the entropic penalty can preferentially smooth distances among delocalized chaotic states, it is unclear whether the observed dimensional reduction is driven by the underlying dynamics or by the numerical scheme; this is load-bearing for the central result.

    Authors: We agree that a systematic robustness check against the regularization parameter is necessary to confirm that the dimensional reduction is not an artifact of the entropic smoothing. In the revised manuscript we will add a scan over regularization strengths spanning an order of magnitude and show that the reported trend persists. We chose the Husimi Q-function for its strict positivity, which guarantees well-defined optimal-transport plans; nevertheless, we will include a short discussion of preliminary Wigner-function calculations that produce qualitatively consistent dimensional reduction, while noting the additional numerical care required to handle negativity. These additions will directly address whether the effect is driven by the dynamics rather than the numerical scheme. revision: yes

  2. Referee: [Abstract] Abstract: the assertion that the Wasserstein distance 'correctly captures the Lyapunov exponent' at the separatrix requires quantitative comparison (e.g., extracted exponent versus classical value, error bars, and dependence on state selection). Without these details the agreement remains qualitative and does not yet establish the method as a reliable quantum Lyapunov diagnostic.

    Authors: We accept that the current presentation is largely qualitative. In the revision we will extract the Lyapunov exponent from the scaling of the Wasserstein distance near the separatrix, report the numerical value together with error bars obtained from an ensemble of state selections, and provide a direct numerical comparison to the known classical Lyapunov exponent of the inverted oscillator. The updated text and supplementary figures will make the quantitative agreement explicit. revision: yes

Circularity Check

0 steps flagged

No significant circularity; results from explicit numerical computations on a concrete model

full rationale

The paper's core claims rest on direct computations: applying Sinkhorn-regularized optimal transport to Husimi Q-representations of energy eigenstates in a quantum coupled harmonic oscillator, then using Gram-spectrum embedding to extract effective dimension. These steps are data-driven and model-specific rather than reducing by construction to fitted parameters, self-definitions, or prior results. The reference to arXiv:2604.17649 is used only to interpret the dimensional reduction as supporting a broader conjecture about emergent holographic space; it does not justify or derive the reported numerical trends. No load-bearing self-citation chains, ansatzes smuggled via citation, or renaming of known results appear in the derivation. The work is self-contained against external benchmarks via its explicit model calculations.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Only the abstract is available, so the ledger is necessarily incomplete. The approach relies on standard quantum phase-space representations and optimal-transport techniques whose validity for chaos diagnostics is taken as given.

axioms (1)
  • domain assumption Husimi Q-representations combined with Sinkhorn-regularized optimal transport distances yield an embedding geometry that faithfully encodes chaotic properties of quantum eigenstates.
    Invoked to justify the construction of the Wasserstein space and the subsequent dimensional analysis.

pith-pipeline@v0.9.0 · 5737 in / 1386 out tokens · 36129 ms · 2026-05-21T04:00:16.674620+00:00 · methodology

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Reference graph

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