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

TL;DR

This paper explores how holographic spacetime can emerge from quantum systems via optimal transport and Wasserstein distances, linking quantum dynamics to spacetime geometry.

Contribution

It demonstrates that the 1-Wasserstein distance between quantum states can produce emergent spacetime structures, connecting optimal transport to holography.

Findings

Wasserstein space derived from Husimi Q-representations forms an emergent space.

Lindblad evolution traces an emergent Wasserstein spacetime with black hole-like properties.

Wasserstein space aligns with AdS${}_2$ black hole geometry in SYK models.

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.