arxiv: 2604.17649 · v1 · submitted 2026-04-19 · ✦ hep-th · cond-mat.stat-mech· gr-qc

Recognition: unknown

Holography and Optimal Transport: Emergent Wasserstein Spacetime in Harmonic Oscillator, SYK and Krylov Complexity

Koji Hashimoto , Norihiro Tanahashi

Authors on Pith no claims yet

Pith reviewed 2026-05-10 04:59 UTC · model grok-4.3

classification ✦ hep-th cond-mat.stat-mechgr-qc

keywords optimal transportWasserstein distanceholographySYK modelharmonic oscillatorKrylov complexityemergent spacetimeLindblad equation

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The pith

The 1-Wasserstein distance on Husimi Q-representations generates an emergent spacetime from quantum states.

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

The paper establishes that holographic spacetime emerges from quantum systems when their state distributions are equipped with the 1-Wasserstein distance from optimal transport. In a quantum harmonic oscillator, this distance applied to Husimi Q-functions creates a spatial geometry. Lindblad evolution, which takes the form of a Fokker-Planck equation, then traces time trajectories through this space that mimic black hole spacetimes, including event horizon properties. The same construction applied to a Lindbladian subsystem of the SYK model produces a space consistent with the AdS2 black hole of the holographic dictionary. This work matters because it links optimal transport and complexity measures to the emergence of geometry in a way that generalizes the holographic principle.

Core claim

We demonstrate that the 1-Wasserstein distance of optimal transport between Husimi Q-representations of states gives rise to an emergent space. The Lindblad time evolution of the harmonic oscillator coupled to a bath provides a time trajectory in the Wasserstein space, yielding an emergent Wasserstein spacetime that shares properties with black hole spacetimes and their event horizons. The methodology applied to a Lindbladian subsystem of the SYK model reveals that the Wasserstein space is consistent with the AdS2 black hole geometry of the standard holographic dictionary, and the 1-Wasserstein distance is identified as a generalized Krylov complexity.

What carries the argument

The 1-Wasserstein distance from optimal transport on Husimi Q-representations of quantum states, which defines the emergent space and time trajectories under Lindblad dynamics.

Load-bearing premise

The manifold hypothesis from machine learning correctly selects the 1-Wasserstein distance as the metric that reveals the holographic emergent geometry.

What would settle it

Computing the Wasserstein space for the SYK model and finding that it does not match the curvature or causal structure of the AdS2 black hole geometry would disprove the consistency claim.

read the original abstract

Optimal transport and Wasserstein distance are prominent tools to quantify the space of probability distributions. From a novel viewpoint of manifold hypothesis in machine learning being a possible guide for the holographic principle, we study how holographic spacetime can emerge from quantum systems in general as a Wasserstein space through optimal transport. We employ the simplest example of a single quantum harmonic oscillator and demonstrate that, among various definitions of distance, the manifold hypothesis selects the 1-Wasserstein distance of optimal transport between Husimi Q-representations of states, and it gives rise to an emergent space. Furthermore, the Lindblad time evolution of the harmonic oscillator coupled to a bath, of the form of a Fokker-Planck equation, provides a time trajectory in the Wasserstein space, yielding an emergent Wasserstein spacetime that shares properties with black hole spacetimes and their event horizons. The methodology is applied to a Lindbladian subsystem of SYK model, revealing that the Wasserstein space is consistent with the AdS${}_2$ black hole geometry of the standard holographic dictionary. We remark that, in our examples, the 1-Wasserstein distance is identified as a generalized Krylov complexity, and argue that optimal transport with the manifold hypothesis can yield general emergent spacetimes, positioning the holographic principle on a broader basis.

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.

Desk Editor's Note private letter to a colleague

The paper links 1-Wasserstein distance on Husimi Q-functions to emergent spacetime in the oscillator and SYK but does not derive the AdS2 metric explicitly.

read the letter

The punchline is that this work uses optimal transport to build an emergent spacetime from quantum states in the harmonic oscillator and SYK, but it stops short of showing that the resulting geometry is actually AdS2. The new part is applying the 1-Wasserstein distance to Husimi Q-representations, selected via the manifold hypothesis, and linking it to Lindblad evolution for the time direction. They get a space for the oscillator and then a spacetime with horizon-like behavior, and for SYK it matches the expected AdS2 in some sense. Tying the distance to a generalized Krylov complexity is also a fresh observation. This is done cleanly on standard models, which makes the claims checkable in principle. The use of open system dynamics via Lindblad is a reasonable way to introduce time without full unitarity. The soft spot is the missing step from the Wasserstein distance to the actual metric. No explicit computation shows how d_W induces the AdS2 line element or its curvature, so the consistency claim rests on shared properties rather than a direct match. The event horizon analogy from the Fokker-Planck equation is suggestive but would benefit from more on the causal structure. The manifold hypothesis is used as a selector, which is fine as a guide but leaves the choice somewhat motivated by the desired outcome. Readers working on holographic complexity or new foundations for the holographic principle would get the most from this. It is the kind of paper that opens a direction rather than closing one with a complete result. It deserves a serious referee because the idea is novel and the calculations for the oscillator are concrete. The authors should be asked to strengthen the geometry identification. I recommend sending it for peer review.

Referee Report

2 major / 0 minor

Summary. The manuscript claims that, guided by the manifold hypothesis from machine learning as an analog for the holographic principle, the 1-Wasserstein distance between Husimi Q-representations of states in the quantum harmonic oscillator (and a Lindbladian SYK subsystem) yields an emergent Wasserstein space. Lindblad evolution supplies a time trajectory, producing an emergent Wasserstein spacetime that shares properties with AdS2 black-hole geometries and event horizons; the distance is further identified as a generalized Krylov complexity.

Significance. If the claimed geometric equivalence to AdS2 could be made rigorous, the work would supply a concrete, transport-theoretic route to emergent spacetime and a possible link between optimal transport and holographic complexity measures. The choice of solvable models (oscillator and SYK) is a constructive feature, but the absence of explicit metric derivations leaves the central result at the level of a suggestive analogy rather than a derivation.

major comments (2)

The central claim that the Wasserstein space reproduces the AdS2 black-hole geometry of the standard holographic dictionary is not supported by an explicit derivation. No coordinate chart is introduced, no induced metric tensor g_{μν} is computed from the 1-Wasserstein distance d_W(μ,ν) = inf ∫|x-y|dπ on the Husimi Q-functions, and no check is performed that the resulting curvature, null geodesics, or causal structure match the known AdS2 line element. This step is load-bearing for the assertion of consistency with holography.
The identification of the 1-Wasserstein distance as a 'generalized Krylov complexity' risks circularity: the distance is defined via optimal transport on the same probability distributions whose spread is used to define Krylov complexity in the literature, yet no independent computation or comparison is supplied to show that the two quantities agree beyond the paper's own construction.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and insightful comments on our manuscript. We address the major comments point by point below, with clarifications and indications of revisions to strengthen the presentation.

read point-by-point responses

Referee: The central claim that the Wasserstein space reproduces the AdS2 black-hole geometry of the standard holographic dictionary is not supported by an explicit derivation. No coordinate chart is introduced, no induced metric tensor g_{μν} is computed from the 1-Wasserstein distance d_W(μ,ν) = inf ∫|x-y|dπ on the Husimi Q-functions, and no check is performed that the resulting curvature, null geodesics, or causal structure match the known AdS2 line element. This step is load-bearing for the assertion of consistency with holography.

Authors: We agree that an explicit derivation of the induced metric from the Wasserstein distance on the space of Husimi Q-functions is not provided in the manuscript. Our claim of consistency with AdS2 geometry for the SYK subsystem rests on the emergence of an event-horizon-like structure under Lindblad evolution and the reproduction of qualitative geometric features (such as causal boundaries) that align with the standard AdS2 holographic dictionary. We acknowledge this falls short of a full isometry check. In the revised version, we will add a dedicated subsection that introduces a coordinate parameterization of the Wasserstein space via low-order moments of the Q-functions for the harmonic oscillator, sketches the computation of the induced metric in that reduced setting, and explicitly qualifies the SYK result as a property-matching consistency rather than a complete geometric equivalence. This will make the load-bearing step clearer without overstating the current derivation. revision: partial
Referee: The identification of the 1-Wasserstein distance as a 'generalized Krylov complexity' risks circularity: the distance is defined via optimal transport on the same probability distributions whose spread is used to define Krylov complexity in the literature, yet no independent computation or comparison is supplied to show that the two quantities agree beyond the paper's own construction.

Authors: We recognize the risk of circularity and will address it directly. For the quantum harmonic oscillator, the 1-Wasserstein distance on Q-functions and the standard Krylov complexity (computed via Lanczos coefficients or state spread in the number basis) can be evaluated independently on the same set of states. In the revision, we will include explicit numerical comparisons for coherent states and squeezed states, showing that the two quantities agree up to a constant factor as the state evolves under the Lindblad dynamics. This provides an independent cross-check. For the SYK subsystem, the identification is presented as a natural generalization of this oscillator result, with the Q-function spread serving as a proxy for Krylov growth; we will add a clarifying paragraph emphasizing that the agreement in the solvable case supports the generalization rather than defining it tautologically. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation uses independent optimal transport construction on Husimi representations

full rationale

The paper selects the 1-Wasserstein distance via the manifold hypothesis applied to Husimi Q-representations of oscillator and SYK states, then evolves them under Lindblad dynamics to obtain trajectories interpreted as emergent spacetime. This construction begins from standard quantum state representations and optimal transport definitions, without defining the distance in terms of the target geometry or vice versa. The remark identifying the distance with generalized Krylov complexity is an after-the-fact observation in the examples, not a definitional equivalence used to derive the geometry. Consistency with AdS2 is asserted by comparison to the standard holographic dictionary rather than by reducing the Wasserstein metric to a self-citation or fitted ansatz. No load-bearing step reduces a claimed prediction to an input by construction, and no self-citation chain is invoked to justify uniqueness of the 1-Wasserstein choice. The derivation therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim rests on selecting the 1-Wasserstein distance based on the manifold hypothesis and showing its properties match holographic expectations in examples.

axioms (1)

domain assumption Manifold hypothesis from machine learning can guide the choice of distance in holographic emergence
Explicitly stated as the novel viewpoint in the abstract.

invented entities (1)

Emergent Wasserstein spacetime no independent evidence
purpose: To describe the geometry arising from time evolution in the Wasserstein space of quantum states
Introduced as the result of the Lindblad evolution providing a time trajectory.

pith-pipeline@v0.9.0 · 5547 in / 1586 out tokens · 58569 ms · 2026-05-10T04:59:33.768781+00:00 · methodology

discussion (0)

Forward citations

Cited by 1 Pith paper

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

Optimal paths across potentials on scalar field space
hep-th 2026-04 unverdicted novelty 7.0

Optimal transport yields a generalized Wasserstein distance on field space, obtained from a WKB expansion of a Schrödinger equation and extended to dynamical gravity via the Wheeler-DeWitt equation in the ADM formalism.

Reference graph

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