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

This paper proposes that holographic spacetime emerges as a Wasserstein space through optimal transport of quantum state distributions, demonstrating that the 1-Wasserstein distance between Husimi Q-representations serves as a generalized Krylov complexity to reproduce black hole geometries in harmonic oscillators and SYK models under the manifold hypothesis.

The Gist

Imagine you are trying to understand the shape of a vast, invisible city. You can't see the streets or buildings directly; you can only see the "shadows" or "projections" of the city cast onto a flat wall. This is a bit like the Holographic Principle in physics: the idea that our 3D universe (with gravity and black holes) might actually be a projection of information living on a lower-dimensional boundary.

The big question has always been: How do we turn that flat, 2D information into a 3D map?

This paper by Koji Hashimoto and Norihiro Tanahashi proposes a new, surprisingly simple way to build that map. They borrow two powerful ideas from Artificial Intelligence and Machine Learning: Optimal Transport and the Manifold Hypothesis.

Here is the story of their discovery, explained simply.

1. The Problem: Too Much Noise, Not Enough Shape

In quantum physics, a system (like an atom) is described by a "wave function." Think of this as a cloud of probability. If you have many different states of this system, you have a massive, infinite-dimensional library of clouds.

To find the "hidden city" (the emergent spacetime), the authors needed to figure out which clouds are close to each other and which are far apart.

• The old way: Measuring the difference between clouds was like trying to compare two clouds by counting how many raindrops are in the exact same spot. It's messy, and it often says two very different clouds are "zero distance" apart if they don't overlap perfectly.
• The new way (Optimal Transport): Imagine you have a pile of sand (a probability distribution) and you want to move it to form a new shape. Optimal Transport asks: "What is the cheapest, most efficient way to move every grain of sand from the first shape to the second?" The "cost" of this move becomes the distance between the two states.

This distance is called the Wasserstein Distance. It's like measuring how much effort it takes to reshape one cloud into another, rather than just looking at them side-by-side.

2. The Discovery: The "Swiss Roll" of Quantum States

The authors used a Manifold Hypothesis. In machine learning, this is the idea that even if data looks like it's floating in a huge, messy 3D room, it's actually just a thin, curved sheet (like a rolled-up Swiss roll) hidden inside that room.

They tested this on a Quantum Harmonic Oscillator (the simplest model of a vibrating particle, like a spring). They asked:

• Which "map" of the quantum state should we use? (The raw probability or a smoothed-out version called the Husimi Q-representation?)
• Which "distance" measure works best? (The 1-Wasserstein, 2-Wasserstein, etc.?)

The Result:
When they used the Husimi Q-representation and the 1-Wasserstein distance (the simplest version of the "moving sand" cost), magic happened.

• All the complex, infinite-dimensional quantum states collapsed down into a single straight line.
• This line wasn't just a random line; it was the Energy Axis. The distance between two states on this line was directly related to their energy difference.
• The Metaphor: Imagine a tangled ball of yarn (the quantum states). The authors found a specific way to pull the thread that instantly straightened it out into a single, perfect string. That string is the "emergent space."

3. Adding Time: The Black Hole Emerges

A space is boring without time. So, they added a "bath" (like putting the spring in a warm room) to make the system evolve over time. This is described by the Lindblad equation.

As the system evolved, the "sand" (the probability distribution) started to flow.

• They tracked how the "distance" (the Wasserstein distance) changed over time.
• They found that the way the distance grew and slowed down looked exactly like a particle falling into a Black Hole.
• The Horizon: As the particle (the quantum state) fell deeper, it seemed to slow down and freeze from the outside observer's perspective. This is the famous "event horizon" effect of a black hole, where time appears to stop.
• The Conclusion: By simply watching how a quantum system relaxes to its ground state, they reconstructed the geometry of a Black Hole spacetime. The "radial direction" of the black hole (how deep you are inside) was literally the "distance" between the quantum states.

4. The SYK Model: A Real-World Check

The harmonic oscillator is a toy model. To see if this works for "real" physics, they applied the same method to the SYK Model, a complex quantum system famous for being a perfect holographic dual to a Black Hole in 2D space (AdS/CFT correspondence).

The Result:
It worked perfectly. The Wasserstein distance in the SYK model mapped exactly onto the geometry of an AdS2 Black Hole. This confirmed that their method isn't just a fluke for simple springs; it might be a fundamental rule of how spacetime emerges from quantum information.

5. The Secret Link: Complexity

Finally, they noticed something profound. The math they used to calculate the "distance" (Wasserstein) was almost identical to a concept called Krylov Complexity, which measures how "complicated" a quantum state gets as it evolves.

They argued that Optimal Transport is the physical realization of Complexity.

• Krylov Complexity asks: "How many steps does it take to build this state?"
• Wasserstein Distance asks: "How much effort does it take to move the sand to make this state?"
• The paper suggests these are the same thing. The "cost" of moving probability is the "complexity" of the universe, and that cost is the geometry of spacetime.

Summary: The Big Picture

This paper suggests that spacetime is not a fundamental stage where physics happens. Instead, spacetime is an emergent map that appears when we look at how quantum states are related to each other.

• The Tool: Optimal Transport (the most efficient way to move probability).
• The Rule: The Manifold Hypothesis (finding the hidden low-dimensional shape).
• The Result: When you measure the "effort" to change a quantum state, you are actually measuring the distance in space. When you watch that effort change over time, you are watching a Black Hole form.

It's like realizing that the map of a city isn't drawn on paper; the map is the traffic patterns. If you understand how efficiently people move from A to B, you can reconstruct the entire city layout, including the traffic jams (black holes) where movement slows to a halt.

Technical Summary

1. Problem Statement

The central challenge in the holographic principle (AdS/CFT correspondence) is understanding how a higher-dimensional curved gravitational spacetime emerges from a lower-dimensional boundary quantum field theory (QFT). Specifically, the authors address the "dimensional reduction" problem:

• The Paradox: Quantum states in a Hilbert space are distributions in an infinite-dimensional space. Holography implies that the bulk geometry is finite-dimensional (e.g., $d+1$ dimensions). How does the infinite-dimensional space of quantum states reduce to a finite-dimensional effective manifold?
• The Gap: Standard distance measures in Hilbert space (e.g., Fubini-Study metric, trace distance) often fail to capture the geometric structure necessary for a bulk interpretation, particularly because they treat orthogonal states as having vanishing or degenerate distances, obscuring the "shape" differences between distributions.
• The Hypothesis: The authors propose that the emergence of spacetime can be understood through the Manifold Hypothesis (data from high-dimensional spaces concentrate on low-dimensional manifolds) combined with Optimal Transport theory. They seek to identify the specific distance measure and state representation that allow the space of quantum states to be embedded into a low-dimensional (ideally 1D) curved manifold, interpretable as a holographic spacetime.

2. Methodology

The authors employ a three-pronged strategy to identify the "dictionary" for holography:

1. Distance Measure ($i$): They utilize $p$-Wasserstein distances ($W_p$) from optimal transport theory. Unlike KL divergence or Hilbert norms, $W_p$ measures the "cost" of transforming one probability distribution into another, providing a metric space that captures shape differences.
2. Representation of States ($ii$): They compare two representations of quantum states:
   • The standard probability density $\rho(x) = |\langle x|\psi\rangle|^2$.
   • The Husimi Q-representation $Q(\alpha) = \frac{1}{\pi}|\langle \alpha|\psi\rangle|^2$, where $|\alpha\rangle$ is a coherent state. The Husimi representation is positive semi-definite and normalized, making it suitable for optimal transport.
3. State Selection ($iii$): They focus on energy eigenstates (for static analysis) and Lindblad time-evolved states (for dynamic/spacetime analysis).

Technical Workflow:

• Embedding Analysis: Using the Manifold Hypothesis, they calculate distance matrices for sets of quantum states and perform Classical Multidimensional Scaling (MDS). They determine the minimal embedding dimension ($D_{min}$) required to represent these distances in Euclidean space. A low $D_{min}$ (ideally 1) indicates a successful holographic reduction.
• Metric Reconstruction: For the optimal choice, they reconstruct the bulk metric by treating the Wasserstein distance as a radial coordinate ($z$) and mapping the time evolution of the distance to the trajectory of a particle falling into a black hole.
• Model Application:
  • Toy Model: Single Quantum Harmonic Oscillator (analytically tractable).
  • Realistic Model: SYK (Sachdev-Ye-Kitaev) model subsystem (known to be dual to AdS$_2$ gravity).

3. Key Contributions and Results

A. Selection of the Optimal Holographic Dictionary (Harmonic Oscillator)

• Result: Through numerical embedding analysis of energy eigenstates ($n=0$ to $N-1$), the authors found that:
  • Using the probability distribution $\rho(x)$ with any $W_p$ fails to yield a low-dimensional embedding (negative eigenvalues in the Gram matrix for $p=1$, high dimensions for $p>1$).
  • Using the Husimi Q-representation with $p=1$ (1-Wasserstein distance) yields an exact 1-dimensional embedding ($D_{min} = 1$).
• Implication: The 1-Wasserstein distance between Husimi Q-representations is the unique choice that satisfies the Manifold Hypothesis for the harmonic oscillator, reducing the infinite-dimensional Hilbert space to a 1D "energy space."

B. Emergent Black Hole Spacetime (Lindblad Dynamics)

• Setup: The harmonic oscillator is coupled to a bath via a Lindblad master equation (jump operator $\hat{a}$), causing the state to relax from an excited state $|m\rangle$ to the ground state $|0\rangle$.
• Dynamics: The time evolution of the 1-Wasserstein distance, $W_1(t)$, behaves like a particle falling in a gravitational potential:
  • Early time: Linear growth ($W_1 \sim t$).
  • Late time: Exponential approach to an asymptotic value ($W_1 \to Z(m)$), mimicking the redshift near a horizon.
• Metric Reconstruction: By identifying $W_1$ as the radial coordinate $z$, the authors reconstruct a metric:
  $$ds^2 = \sigma(z) \left( -f(z)dt^2 + \frac{1}{f(z)}dz^2 \right)$$
  where $f(z) \sim (z^* - z)$ near the horizon $z^*$. This is the metric of a black hole with an event horizon.
• Consistency: Multiple trajectories starting from different initial energies $m$ all converge to the same null geodesic structure in the emergent spacetime, confirming a consistent bulk geometry.

C. Validation with the SYK Model

• Setup: A two-Majorana-fermion subsystem of the SYK model is treated as a Lindblad system coupled to the rest of the system (the bath).
• Result: The time evolution of the 1-Wasserstein distance in the SYK subsystem follows $W_1(t) \propto (1 - e^{-cTt})$.
• Significance: This functional form exactly matches the trajectory of a null shock wave falling into an AdS$_2$ Schwarzschild black hole geometry. This provides a non-trivial consistency check, linking the optimal transport approach to the standard AdS/CFT dictionary.

D. Connection to Krylov Complexity

• Discovery: The authors demonstrate that the 1-Wasserstein distance for the Lindblad harmonic oscillator is mathematically equivalent to a generalized Krylov complexity.
• Mechanism:
  • Standard Krylov complexity $K(t) = \sum n |\phi_n(t)|^2$ counts the "spread" of an operator in a Krylov basis.
  • The 1-Wasserstein distance takes the form $W_1(t) = \sum C_k(t) [Z(m) - Z(k)]$.
  • Here, the "cost" $Z(m) - Z(k)$ (derived from the Wasserstein coordinate) replaces the integer index $n$.
• Wasserstein Operator: They define a "Wasserstein operator" $\hat{W}$ such that $W_1(t) = \langle \psi(t) | \hat{W} | \psi(t) \rangle$. This operator effectively measures the energy (or square root of energy) in the coherent state basis.
• Advantage: Unlike standard Krylov complexity which depends on the initial operator choice, the 1-Wasserstein distance defines a metric space independent of the initial state, allowing for universal comparisons of complexity growth.

4. Significance and Conclusion

• New Perspective on Holography: The paper positions the holographic principle not just as a duality of operators, but as a dimensional reduction mechanism driven by optimal transport. It suggests that the "bulk" geometry is the intrinsic low-dimensional manifold of probability distributions (specifically Husimi Q-distributions) equipped with the 1-Wasserstein metric.
• Resolution of Dimensionality: It resolves the issue of infinite-dimensional Hilbert spaces by showing that for specific physical states and representations, the effective degrees of freedom collapse to a 1D manifold (energy space).
• Black Hole Emergence: It provides a concrete mechanism where the thermodynamic relaxation of a quantum system (Lindblad evolution) naturally generates a spacetime metric with an event horizon, offering a microscopic derivation of black hole redshift and horizon properties.
• Bridge to Machine Learning: By leveraging the Manifold Hypothesis and Optimal Transport (tools from AI/ML), the paper bridges the gap between quantum information theory, machine learning, and high-energy physics, suggesting that the geometry of data (quantum states) is the fundamental origin of spacetime.

In summary, Hashimoto and Tanahashi argue that 1-Wasserstein distance on Husimi Q-representations is the correct geometric tool to reconstruct holographic spacetime, successfully reproducing black hole geometries in both toy models and the SYK model, while identifying this distance as a generalized form of Krylov complexity.