Holography and Optimal Transport: Emergent Wasserstein Spacetime in Harmonic Oscillator, SYK and Krylov Complexity
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2026
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| author | Hashimoto, Koji Tanahashi, Norihiro |
| author_facet | Hashimoto, Koji Tanahashi, Norihiro |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_17649 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Holography and Optimal Transport: Emergent Wasserstein Spacetime in Harmonic Oscillator, SYK and Krylov Complexity Hashimoto, Koji Tanahashi, Norihiro High Energy Physics - Theory Statistical Mechanics General Relativity and Quantum Cosmology 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. |
| title | Holography and Optimal Transport: Emergent Wasserstein Spacetime in Harmonic Oscillator, SYK and Krylov Complexity |
| topic | High Energy Physics - Theory Statistical Mechanics General Relativity and Quantum Cosmology |
| url | https://arxiv.org/abs/2604.17649 |