Homological Classification of Few–Qubit Entanglement

This work introduces a homological framework for classifying entanglement in few-qubit quantum systems. By mapping multipartite entangled states to simplicial complexes whose vertices correspond to qubits and whose simplices encode inseparable correlations, the paper defines entanglement complexes whose Betti numbers act as topological invariants. These invariants reproduce the known SLOCC classes — product, Bell, GHZ, and W — for two, three, and four qubits. The approach establishes a direct correspondence between quantum entanglement and low-dimensional topology, framing entanglement connectivity in terms of homology groups and the Euler characteristic. The method distinguishes GHZ-type and W-type entanglement through topological connectivity and provides a foundation for a general entanglement homology theory. Connections are drawn to topological quantum field theory, holographic entanglement geometry, and topological data analysis, suggesting a unified topological language for quantum information. Future work includes developing a functorial cohomological formulation and applying persistent homology to mixed and evolving states.