On the quantum separability of qubit registers

TL;DR

This paper explores how the combinatorial structure of a pure qubit state's support determines its separability, introducing a geometric framework that aids in entanglement diagnostics and quantum circuit analysis.

Contribution

It introduces a support-based taxonomy using Boolean cube geometry to distinguish separable and entangled pure qubit states, providing closed-form support counts and forbidden configurations.

Findings

Support geometry determines bipartite separability.

Forbidden configurations enforce multipartite entanglement.

Framework enables fast entanglement diagnostics.

Abstract

We show that the bipartite separability of a pure qubit state hinges critically on the combinatorial structure of its computational-basis support. Using Boolean cube geometry, we introduce a taxonomy that distinguishes support-guaranteed separability from cases in which entanglement depends on probability amplitudes. We provide closed-form support counts, identify forbidden configurations that enforce multipartite entanglement, and show how these results can enable fast entanglement diagnostics in quantum circuits. The framework offers immediate utility in classical simulation, entanglement-aware circuit design, and quantum error-correcting code analysis. This establishes support geometry as a practical and scalable tool for understanding entanglement in quantum information processing.