Quantum information meets algebraic geometry: A python framework for CES and UPBs

¹ School of Advanced Sciences, Vellore Institute of Technology, Vellore 632014, India
² Automotive Research Centre, Vellore Institute of Technology, Vellore 632014, India

^* Corresponding author: This email address is being protected from spambots. You need JavaScript enabled to view it.

Published online: 23 March 2026

Abstract

The present paper introduces a Python-based framework customized for the structured exploration of subspaces in C_m ⊗ C_n that do not contain any product states-known as completely entangled subspaces (CES). The frame makes use of recent developments in quantum information and algebraic geometry to enable the construction, verification, and geometric characterization of CES, supporting both theoretical and numerical studies. By integrating algorithms for entanglement detection and geometric quantification, the tool enables exploration of maximal dimension CES and their relationships with unextendible product bases (UPBs). The approach utilizes totally non-singular matrices, such as Vandermonde matrices, to generate non-orthogonal product bases and solve homogeneous systems of linear equations for basis vector determination. The implementation supports efficient manipulation of quantum operators and their ranges, permit recognition of genuinely entangled multipartite subspaces derived from bipartite systems. The framework is designed for scalability to higher-dimensional and multipartite framework, providing a flexible platform for researchers investigating the geometry of entanglement and separability in quantum systems. Contributions include the incorporation of quantum support vector machines and classical deep neural networks for entanglement detection, the use of entanglement witnesses for verification, and advanced visualization tools for geometric properties. The framework draws on foundational results and recent advances in quantum information, offering a robust and accessible resource for both theoretical and practical research in quantum entanglement. This work advances the field by providing an expandable and adaptable tool for studying CES in C^m ⊗ Cⁿ