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.
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.
Genuine multipartite entanglement (GME) represents the strongest form of quantum correlation, where all parties of a system across each bipartition are inseparably linked. This resource is crucial for advanced quantum protocols, including multi-party cryptography and distributed quantum computing, yet its quantification remains a significant theoretical challenge. This review surveys a rapidly evolving geometric paradigm that translates this challenge into a problem of polygon inequalities. By representing marginal bipartite entanglements as side lengths, the distribution of entanglement in a multipartite state must satisfy geometric closure conditions akin to forming a polygon. These polygon-based constraints provide a unifying framework for detecting and quantifying GME, offering an intuitive bridge between bipartite entanglement measures and global multipartite structure. We chart the development of this approach from its monogamy-based origins to modern formulations using concurrence triangles, wedge products, and entropy-based polygons. Recent advances have established rigorous conditions for these geometric quantities to serve as faithful, operationally meaningful entanglement measures, while also extending the framework to mixed states and higher-dimensional systems. Despite progress, key challenges persist in ensuring universal monotonicity and experimental robustness. By synthesizing these geometric insights, this review highlights how polygon inequalities are reshaping the quantification of complex quantum correlations and providing novel diagnostic tools for the next generation of quantum technologies.
Determining whether a subspace spanned by certain quantum states is entangled and its entanglement dimensionality remains a fundamental challenge in quantum information science. This paper introduces a geometric measure of r-bounded rank, Er(S), for a given subspace S. Derived from the established geometric measure of entanglement, this measure is specifically designed to assess the entanglement within S. It not only serves as a tool for determining the entanglement dimensionality but also illuminates the subspace's capacity to preserve such entanglement. By employing developed nonconvex optimization techniques utilized in machine learning area, we can accurately calculate Er(S) within the manifold optimization framework. Our approach demonstrates notable advantages over existing hierarchical methods, PPT relaxation techniques, and the seesaw strategy, particularly by combining computational efficiency with broad applicability. More importantly, it paves the way for high-dimensional entanglement certification, which is crucial for numerous quantum information tasks. We showcase its effectiveness in validating high-dimensional entangled subspaces in bipartite systems, determining the border rank of multipartite pure states, and identifying genuinely or completely entangled subspaces.
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.
In the studies of quantum information and computation, the quantum correlations
present in a non-classical system are seen as natural resources. In this
thesis, we present a detailed study on quantification and characterization of
multipartite quantum entanglement from a geometric perspective. Furthermore,
we present protocols and algorithms where quantum correlations such
as entanglement, quantum Fisher information, and quantum superposition
have been used to provide advantages as well securities in quantum communication
tasks, quantum speed-up, and protection against the decoherence in
an open quantum system.
First, we use a geometric measure of entanglement quantification based
on Euclidean distance of the Hermitian matrices to obtain the minimum distance
between the set of bipartite n- qudit density matrices with a positive
partial transpose and the maximally mixed state. This minimum distance coincides
with the radius of the largest separable ball. We present the interior
geometry of the set of all positive semidefinite matrices and identify a particular
class of Werner states for which the Peres-Horodecki criterion is both
necessary and sufficient for separability even in the dimensions greater than
six.
ABSTRACT Characterizing the entanglement in multipartite quantum systems remains a central challenge in the theory of quantum entanglement. We establish a comprehensive framework for constructing ‐entanglement measures by leveraging bipartite entanglement measures and general ‐entanglement measures () with arithmetic and geometric averaging schemes. Numerical examples demonstrate that these measures outperform the existing ones, such as ‐ME concurrence, geometric mean (k‐GM) concurrence, and the generalized ‐entanglement measure by inducing distinct entanglement orderings and offering superior resolution. Furthermore, we reveal the operational relationship between the ‐entanglement and the local quantum coherence, verifying that the persistence of entanglement is intrinsically constrained by the system's capacity to maintain the local coherence. Finally, to overcome the inherent limitation of the conventional three‐tangle in characterizing the genuine multipartite entanglement of the GHZ‐W mixed states, we design a novel GME measure. Numerical results confirm that this measure effectively eliminates the detection blind spots where vanishes, providing a complete and practical mathematical tool for the quantitative analysis of genuine multipartite quantum correlations.
Entanglement, a fundamental resource in quantum information science, underpins many quantum technologies, including quantum computation and cryptography. Quantifying entanglement is a critical challenge, especially for mixed states and higher-dimensional systems, where existing measures often face computational complexities or lack analytical closed-form expressions. This paper introduces a novel approach to entanglement quantification utilizing determinantal polynomials. We propose a framework where entanglement properties of quantum states, particularly their degree and type, can be characterized and potentially measured through the analysis of determinants derived from state representations, such as density matrices and their partial transposes. By constructing specific polynomial functions whose roots or coefficients are directly linked to entanglement measures or criteria, we aim to provide a more tractable and potentially analytical pathway for quantifying entanglement. The methodology leverages advanced concepts from linear algebra and algebraic geometry, connecting the algebraic properties of quantum states to their entanglement features. This work discusses the theoretical underpinnings, provides illustrative examples for low-dimensional systems, and explores the potential of this method to overcome some limitations of current entanglement measures, offering a new perspective on the algebraic structure of entanglement.