Wasserstein Space of Energy Eigenstates

• The Wasserstein space of energy eigenstates is a phase-space construction where quantum states are represented by Husimi probability distributions with optimal-transport distances.
• In the harmonic oscillator, discrete energy levels form a one-dimensional metric space defined by the monotone Wasserstein coordinate Z(n) that asymptotically grows as √n.
• Extensions to dissipative dynamics and chaotic systems provide a diagnostic framework linking emergent spacetime geometry, scrambling, and scar structure.

The Wasserstein space of energy eigenstates is a metric construction in which each energy eigenstate is replaced by a probability distribution—typically a Husimi $Q$-distribution on phase space—and pairwise distances are defined by optimal transport rather than by a unitarily invariant Hilbert-space metric. In the harmonic-oscillator realization, this procedure turns the discrete family $\{|n\rangle\}$ into an exactly one-dimensional metric space under the $1$-Wasserstein distance, with a monotone coordinate $Z(n)$ that grows asymptotically as $\sqrt{n}$. In the broader program developed in recent work, this emergent geometry is interpreted as an “energy space,” extended to dissipative trajectories as an emergent Wasserstein spacetime, and used as a diagnostic framework for chaos, scrambling, and scar structure in more complicated quantum systems (Hashimoto et al., 19 Apr 2026).

1. Definition and conceptual setting

In the usage established by recent work, the Wasserstein space of energy eigenstates is the metric space obtained by mapping selected energy eigenstates $\{|n\rangle\}$ to phase-space probability distributions and then equipping that set with pairwise Wasserstein optimal-transport distances. The central construction is explicitly basis-dependent in a controlled sense: it is not a geometry of Hilbert-space rays under abstract distances such as Fubini–Study, trace, or Bures, but a geometry of semiclassical phase-space distributions. This controlled basis dependence is essential because the intended geometric content concerns energy shells, separatrices, regular islands, scars, and related phase-space structures (Hashimoto et al., 20 May 2026).

For the harmonic oscillator, the preferred probability representation is the Husimi $Q$-representation rather than the position-space density alone. For a pure state $|\psi\rangle$, the two positive normalized distributions considered are

$$\rho(x)=|\langle x|\psi\rangle|^2, \qquad Q(\alpha)=\frac{1}{\pi}|\langle \alpha|\psi\rangle|^2,$$

where $|\alpha\rangle$ is a coherent state. The phase-space label $\{|n\rangle\}$0 is interpreted semiclassically as the oscillator phase-space coordinate. The Husimi distribution is geometrically natural, positive semidefinite, and normalized, unlike the Wigner function, and the positivity is explicitly emphasized as the reason the Husimi representation is preferred for defining Wasserstein distances (Hashimoto et al., 19 Apr 2026).

This construction differs sharply from other quantum optimal-transport programs. In "Geometry of Grassmannians and optimal transport of quantum states" (Antonini et al., 2021), density matrices are seen as discrete probability measures on the Grassmannian $\{|n\rangle\}$1 via the spectral theorem, so the basic geometric points are spectral subspaces rather than phase-space Husimi distributions. The Wasserstein space of energy eigenstates in the phase-space sense therefore concerns the transport geometry of eigenstate-associated probability densities, not the Grassmannian geometry of eigenspaces themselves (Antonini et al., 2021).

2. Harmonic-oscillator realization

The most explicit realization is the single quantum harmonic oscillator with

$\{|n\rangle\}$2

Its energy eigenstate wavefunctions are

$\{|n\rangle\}$3

with position-space probability density $\{|n\rangle\}$4. More important for the construction is the Husimi distribution

$\{|n\rangle\}$5

which becomes

$\{|n\rangle\}$6

Because $\{|n\rangle\}$7 depends only on $\{|n\rangle\}$8, it is rotationally symmetric, and the problem reduces to the one-dimensional radial distribution

$\{|n\rangle\}$9

The paper notes that this is a Gamma distribution, so each eigenstate is represented by a radial probability profile in phase space, peaked at larger $1$0 as $1$1 increases (Hashimoto et al., 19 Apr 2026).

The Wasserstein family used is the $1$2-Wasserstein distance. In one dimension the paper writes

$1$3

with $1$4 constrained by the usual marginal conditions. For one-dimensional distributions it uses the quantile formula

$1$5

and for $1$6,

$1$7

The paper also invokes the Kantorovich–Rubinstein duality for $1$8 (Hashimoto et al., 19 Apr 2026).

A central claim is that the manifold hypothesis selects the pair consisting of the Husimi $1$9-representation and the $Z(n)$0-Wasserstein distance. The criterion is whether the finite set of states $Z(n)$1, equipped with pairwise distances, embeds into a low-dimensional Euclidean space. For position-space probabilities $Z(n)$2, $Z(n)$3 is not even Euclidean-embeddable and the other $Z(n)$4 need $Z(n)$5, so no robust low-dimensional manifold emerges. For Husimi $Z(n)$6-distributions, $Z(n)$7 is singled out because only it produces an exact one-dimensional geometry for the eigenstates, whereas $Z(n)$8 gives only an approximate one-dimensional structure and $Z(n)$9 fails Euclidean embeddability (Hashimoto et al., 19 Apr 2026).

3. Exact one-dimensional geometry and the Wasserstein coordinate

For harmonic-oscillator eigenstates, the radial CDFs of $\sqrt{n}$0 are completely ordered: they do not intersect. This yields an exact additive form for the $\sqrt{n}$1-Wasserstein distance. For $\sqrt{n}$2,

$\sqrt{n}$3

where the paper defines the “Wasserstein coordinate”

$\sqrt{n}$4

normalized so that $\sqrt{n}$5. Using the explicit $\sqrt{n}$6, this becomes

$\sqrt{n}$7

Therefore

$\sqrt{n}$8

This is the precise construction of the Wasserstein space of energy eigenstates in the harmonic-oscillator case: the discrete set $\sqrt{n}$9 becomes a one-dimensional metric space with coordinate $\{|n\rangle\}$0, and the geometry is exactly a discretized line rather than a merely approximate radial arrangement (Hashimoto et al., 19 Apr 2026).

The asymptotics give the “energy-space” interpretation. Using Gamma-function asymptotics,

$\{|n\rangle\}$1

so

$\{|n\rangle\}$2

Semiclassically,

$\{|n\rangle\}$3

hence $\{|n\rangle\}$4 is interpreted as an energy coordinate in the sense that higher energy eigenstates lie further out along a radial direction whose coordinate grows like the square root of energy. In this sense, the emergent Wasserstein space is a radial geometry organized by energy shells in phase space (Hashimoto et al., 19 Apr 2026).

The paper also gives a continuum metric for this emergent line. Extending the discrete label $\{|n\rangle\}$5 to a continuous coordinate $\{|n\rangle\}$6, it imposes

$\{|n\rangle\}$7

so that

$\{|n\rangle\}$8

At large $\{|n\rangle\}$9,

$Q$0

The Kantorovich–Rubinstein duality provides a second interpretation: because the CDFs are ordered, the optimal dual observable is $Q$1, so the Wasserstein coordinate is the expectation value of the phase-space radius. The paper stresses that this state-independent observable interpretation exists only because of the ordered-CDF property (Hashimoto et al., 19 Apr 2026).

4. Dissipative trajectories, emergent spacetime, and generalized Krylov complexity

The construction extends from pure eigenstates to mixed decohered states under Lindblad evolution. With jump operator $Q$2, the density matrix evolves by

$Q$3

and for diagonal states

$Q$4

Starting from $Q$5, the coefficients are

$Q$6

The corresponding Husimi radial distribution is

$Q$7

Because the CDFs remain ordered in time, the $Q$8-Wasserstein distance from the initial state is

$Q$9

Time evolution therefore traces a curve through the same Wasserstein space whose distinguished points include the energy eigenstates (Hashimoto et al., 19 Apr 2026).

Interpreting the radial bulk coordinate as $|\psi\rangle$0, the paper reconstructs a $|\psi\rangle$1-dimensional emergent spacetime metric from the null-trajectory relation $|\psi\rangle$2. For a fixed initial eigenstate $|\psi\rangle$3, the late-time behavior near the endpoint $|\psi\rangle$4 yields

$|\psi\rangle$5

which is a black-hole horizon form. The eigenstates form the static radial skeleton of the emergent space, while Lindblad evolution produces dynamical trajectories through that space, interpreted as infall toward a horizon. The paper applies the same methodology to a Lindbladian subsystem of the SYK model and remarks that the Wasserstein space is consistent with the AdS$|\psi\rangle$6 black hole geometry of the standard holographic dictionary (Hashimoto et al., 19 Apr 2026).

The same structure is related to Krylov complexity. For diagonal Lindblad evolution,

$|\psi\rangle$7

and since $|\psi\rangle$8, this has the same structure as a generalized Krylov complexity

$|\psi\rangle$9

except that the linear label $$\rho(x)=|\langle x|\psi\rangle|^2, \qquad Q(\alpha)=\frac{1}{\pi}|\langle \alpha|\psi\rangle|^2,$$0 is replaced by the Wasserstein coordinate $$\rho(x)=|\langle x|\psi\rangle|^2, \qquad Q(\alpha)=\frac{1}{\pi}|\langle \alpha|\psi\rangle|^2,$$1. Because $$\rho(x)=|\langle x|\psi\rangle|^2, \qquad Q(\alpha)=\frac{1}{\pi}|\langle \alpha|\psi\rangle|^2,$$2, the $$\rho(x)=|\langle x|\psi\rangle|^2, \qquad Q(\alpha)=\frac{1}{\pi}|\langle \alpha|\psi\rangle|^2,$$3-Wasserstein distance acts like a generalized Krylov complexity weighted by square-root energy or radial position rather than by step number alone. The paper further defines a “Wasserstein operator”

$$\rho(x)=|\langle x|\psi\rangle|^2, \qquad Q(\alpha)=\frac{1}{\pi}|\langle \alpha|\psi\rangle|^2,$$4

for rotationally symmetric Husimi states, so that the $$\rho(x)=|\langle x|\psi\rangle|^2, \qquad Q(\alpha)=\frac{1}{\pi}|\langle \alpha|\psi\rangle|^2,$$5-Wasserstein distance is the change in its expectation value (Hashimoto et al., 19 Apr 2026).

5. Chaos, scrambling, folds, and branches

A broader notion of Wasserstein space of energy eigenstates is developed for nonintegrable systems by mapping each eigenstate to its Husimi $$\rho(x)=|\langle x|\psi\rangle|^2, \qquad Q(\alpha)=\frac{1}{\pi}|\langle \alpha|\psi\rangle|^2,$$6-distribution and computing pairwise optimal-transport distances on a discretized phase-space grid, typically with entropic regularization. The resulting pairwise distance matrix is then converted into an emergent embedding geometry through the centered Gram matrix

$$\rho(x)=|\langle x|\psi\rangle|^2, \qquad Q(\alpha)=\frac{1}{\pi}|\langle \alpha|\psi\rangle|^2,$$7

The eigenvalues of $$\rho(x)=|\langle x|\psi\rangle|^2, \qquad Q(\alpha)=\frac{1}{\pi}|\langle \alpha|\psi\rangle|^2,$$8 furnish a spectral notion of effective dimension: large leading eigenvalues followed by rapid decay indicate that the point cloud lies close to a low-dimensional manifold in the embedding space (Hashimoto et al., 20 May 2026).

In the two-dimensional coupled harmonic oscillator with

$$\rho(x)=|\langle x|\psi\rangle|^2, \qquad Q(\alpha)=\frac{1}{\pi}|\langle \alpha|\psi\rangle|^2,$$9

the paper finds that the effective dimension of the Wasserstein space of low-lying energy eigenstates decreases as the system becomes more chaotic. Fitting the leading Gram eigenvalues to $|\alpha\rangle$0, the baseline calculation gives

$|\alpha\rangle$1

Thus increasing chaoticity corresponds to stronger spectral compression and fewer dominant embedding directions. The paper calls this chaotic dimensional reduction and presents it as support for the conjecture that the Wasserstein space serves as an emergent holographic space through the manifold hypothesis (Hashimoto et al., 20 May 2026).

The same framework is used to analyze scrambling and separatrices in one-dimensional models. The microcanonical OTOC

$|\alpha\rangle$2

identifies states near a saddle that display exponential growth. In the double-well and partly flattened oscillator, precisely those scrambling states appear as fold points of the embedded eigenstate curve. The paper’s conceptual claim is that scrambling induces folding in Wasserstein space and that repeated or cumulative versions of such folding in higher-dimensional, nonintegrable systems may underlie chaotic dimensional reduction. The wording is cautious: folding “may underlie” the reduction, rather than constituting a formal theorem (Hashimoto et al., 20 May 2026).

In the analytically solvable inverted harmonic oscillator,

$|\alpha\rangle$3

Wasserstein distance reproduces the Lyapunov exponent at late time. For $|\alpha\rangle$4-Wasserstein, the asymptotic growth is

$|\alpha\rangle$5

while for $|\alpha\rangle$6-Wasserstein the late-time behavior is $|\alpha\rangle$7. The paper presents this as one of its strongest analytic results: Wasserstein distance captures the quantum Lyapunov exponent at the scrambling point. In the strongly chaotic regime of the coupled oscillator, the embedding also exhibits a distinct branch of states interpreted cautiously as scar-like states, so that folds, dimensional compression, and branches become geometric signatures of scrambling, chaos, and scars, respectively (Hashimoto et al., 20 May 2026).

6. Related formulations, misconceptions, and limitations

Several adjacent constructions clarify what the Wasserstein space of energy eigenstates is and is not. Halliwell’s "Exact Phase Space Localized Projectors from Energy Eigenstates" (Halliwell, 2012) does not define Wasserstein metrics, but it provides a closely related phase-space organization of oscillator energy eigenstates. In that construction, the quasi-projector onto a circular region is diagonal in the number basis,

$|\alpha\rangle$8

and the exact projector becomes

$|\alpha\rangle$9

The paper repeatedly interprets $\{|n\rangle\}$00 as concentrated inside the circle corresponding to its classical energy orbit. This does not amount to a transport geometry, but it supports the broader phase-space view that energy eigenstates are organized by radial energy shells (Halliwell, 2012).

A common misconception is to identify the Wasserstein space of energy eigenstates with a basis-independent quantum-state geometry. The recent phase-space constructions explicitly reject that interpretation. The relevant geometry is built from Husimi $\{|n\rangle\}$01-functions and Euclidean ground cost on phase space, so it is not a geometry of Hilbert-space rays under an abstract unitarily invariant metric. Another misconception is that the exact one-dimensional line found in the harmonic oscillator should apply to the full Hilbert space. The caveat is explicit: the exact one-dimensionality applies to the selected subset of states—harmonic-oscillator energy eigenstates, and more generally decohered mixtures diagonal in that basis with ordered Husimi CDFs—not automatically to the full Hilbert space (Hashimoto et al., 19 Apr 2026).

The technical limitations are equally explicit. The harmonic-oscillator result depends crucially on choosing the Husimi $\{|n\rangle\}$02-representation rather than, for example, the Wigner function; positivity and ordered radial CDFs are essential. The Lindblad spacetime interpretation is described as a model-dependent emergent geometry rather than a literal gravity dual for the single oscillator. In the chaos analysis, the distances are mostly entropically regularized $\{|n\rangle\}$03-Wasserstein distances computed on finite grids, the Euclidean embeddings are approximate because small negative Gram eigenvalues appear after discretization and regularization, and the effective dimension is a spectral diagnostic rather than a rigorous intrinsic dimension in a mathematical sense (Hashimoto et al., 19 Apr 2026, Hashimoto et al., 20 May 2026).

A further extension is suggested by the iterated Wasserstein geometry of random measures developed in "Nested superposition principle for random measures and the geometry of the Wasserstein on Wasserstein space" (Pinzi et al., 8 Oct 2025). That paper does not address quantum eigenstates directly, but it provides a metric geometry on

$\{|n\rangle\}$04

the space of laws of random probability measures. This suggests a natural outer geometry for ensembles of eigenstate-associated Husimi distributions or disorder-averaged families of state distributions, once the single-eigenstate encoding into $\{|n\rangle\}$05 has been fixed (Pinzi et al., 8 Oct 2025).

In its sharpest current form, the Wasserstein space of energy eigenstates is therefore the phase-space transport geometry obtained by encoding each energy eigenstate as a Husimi probability distribution and measuring pairwise optimal-transport costs. For the harmonic oscillator, this yields an exact one-dimensional metric line with coordinate

$\{|n\rangle\}$06

asymptotically $\{|n\rangle\}$07, and a continuum metric satisfying $\{|n\rangle\}$08. In more general systems, the same construction produces approximate embedding geometries whose folds, dimensional compression, and branches encode scrambling, chaos, and scar structure (Hashimoto et al., 19 Apr 2026, Hashimoto et al., 20 May 2026).