Entropy-Topology Correspondence in Quantum Chaotic Systems: A Metric-Preserving Learning Approach to the SYK Model

We investigate the emergent geometry of quantum information manifolds using a metric-preserving machine learning framework. Reconstructing the state manifold of the Sachdev-Ye-Kitaev (SYK) model via a quantum autoencoder constrained by the Quantum Bures Metric, we ensure the latent space preserves the physical distinguishability of states. Our analysis reveals a robust entropytopology correspondence: a strong negative correlation (r ≈ −0.93, extrapolating to r ≈ −1.03 in the thermodynamic limit) between thermodynamic entropy and the box-counting dimension of the manifold. Finite-size scaling (FSS) analysis, performed via linear extrapolation of r as a function of 1/N, indicates this anti-correlation approaches unity in the thermodynamic limit. Crucially, we demonstrate that this geometric smoothing is chaos-dependent: integrable free fermion systems and local Heisenberg chains fail to exhibit the same correspondence. These results provide a geometric signature of quantum scrambling and suggest that maximal chaos is a prerequisite for the emergence of smooth information manifolds in holographic systems.