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 ⊗ C n
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 ⊗ C n
The present paper introduces a Python-based framework customized for the structured exploration of subspaces in Cm ⊗ Cn 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 Cm ⊗ Cn
We propose an algebraic geometry-inspired approach for constructing entangled subspaces within the Hilbert space of a multipartite quantum system. Specifically, our method employs a modified Veronese embedding, restricted to the conic, to define subspaces within the symmetric part of the Hilbert space. By utilizing this technique, we construct the minimal-dimensional, non-orthogonal yet Unextendible Product Basis (nUPB), enabling the decomposition of the multipartite Hilbert space into a two-dimensional subspace, complemented by a Genuinely Entangled Subspace (GES) and a maximal-dimensional Completely Entangled Subspace (CES). In multiqudit systems, we determine the maximum achievable dimension of a symmetric GES and demonstrate its realization through this construction. Furthermore, we systematically investigate the transition from the conventional Veronese embedding to the modified one by imposing various constraints on the affine coordinates, which, in turn, increases the CES dimension while reducing that of the GES.
Constructing optimal entanglement witnesses for arbitrary quantum systems remains a formidable challenge, hindered by the structural complexity of high-dimensional quantum state spaces. In this work, we overcome this barrier by introducing a unified framework for constructing geometrically optimal witnesses capable of identifying arbitrary entanglement structures. By leveraging the geometric properties of a target quantum state relative to a specific entanglement structure in Hilbert-Schmidt space, these witnesses are constructed in an efficient and effective manner. Among all nontrivial witnesses, they provide maximal distinguishability between the target state and the set of states without the desired entanglement structure. As a result, our approach enables the universal detection of bipartite, multipartite, and high-dimensional entanglement. To demonstrate its experimental feasibility, we apply the framework to a four-qubit nuclear spin quantum system. Compared to existing methods, our approach offers enhanced noise robustness, broader detection universality, and improved construction efficiency, providing a powerful and versatile tool for advancing the characterization and understanding of entanglement in quantum systems.
Quantifying quantum entanglement is a pivotal challenge in quantum information science, particularly for high-dimensional systems, due to its computational complexity. This thesis extends the geometric measure of entanglement (GME) to introduce and investigate a suite of GME-based entanglement monotones tailored for diverse quantum contexts, including pure states, subspaces, and mixed states. These monotones are applicable to both bipartite and multipartite systems, offering a unified framework for characterizing entanglement across various scenarios. Notably, the proposed monotones are adept at identifying entanglement with varying entanglement dimensionalities, making them particularly effective for detecting high-dimensional entanglement. To support practical computation, we develop a non-convex optimization framework that yields accurate upper bounds, complemented by semidefinite programming techniques to establish robust lower bounds. Together, these approaches provide a consistent and efficient computational methodology. This work advances both the theoretical understanding and algorithmic tools for entanglement quantification, contributing to the study of complex quantum correlations in entangled systems.
Unextendible product bases (UPBs) are interesting mathematical objects arising in composite Hilbert spaces that have found various applications in quantum information theory, for instance in a construction of bound entangled states or Bell inequalities without quantum violation. They are closely related to another important notion, completely entangled subspaces (CESs), which are those that do not contain any fully separable pure state. Among CESs one finds a class of subspaces in which all vectors are not only entangled but genuinely entangled. Here we explore the connection between UPBs and such genuinely entangled subspaces (GESs) and provide classes of nonorthogonal UPBs that lead to GESs for any number of parties and local dimensions. We then show how these subspaces can be immediately utilized for a simple general construction of genuinely entangled states in any such multipartite scenario.
Unextendible product bases (UPBs) are important structures in quantum information theory, with applications to completely entangled subspaces, bound entanglement, and local indistinguishability. Since many properties and applications of UPBs are closely related to their cardinalities, one of the central problems in the study of UPBs is to determine whether UPBs of prescribed sizes exist in a given multipartite system. In this paper, we introduce a SAT-assisted framework based on decompositions of the N-dimensional hypercube. We define ON-tile decompositions and prove a tile-to-UPB theorem: every ON-tile decomposition induces a UPB through a construction based on tile-wise Fourier product bases and a global stopper state. We then encode the search for such decompositions as a Boolean satisfiability (SAT) problem and use SAT solvers to generate explicit instances. In terms of verification, we also implement a UPB verification algorithm based on local orthogonality graphs and unsaturated subspaces. The algorithm can be used to determine whether an arbitrary finite set of product states forms a UPB. Using this framework, we obtain UPBs of several sizes in some tripartite and quadripartite systems, including sizes 13,14,…,23 in C3⊗C3⊗C3. Moreover, the small-dimensional instances obtained here can serve as seed UPBs for recursive constructions, leading to further examples in larger multipartite systems.