Quantum entanglement is a foundational resource in quantum information science, underpinning applications across physics. However, detecting and quantifying entanglement remains a significant challenge. In this article, we introduce a variational quantum algorithm inspired by Uhlmann's theorem to quantify the Bures entanglement of general quantum states, a method that naturally extends to other quantum resources, including genuine multipartite entanglement, quantum discord, quantum coherence, and total correlations, while also enabling reconstruction of the closest free states. The algorithm requires a polynomial number of ancillary qubits and circuit depth relative to the system size, dimensionality, and free-state cardinality, making it scalable for practical implementations. Thus, it provides a versatile framework for quantifying quantum resources, demonstrated here through several applications.
Quantum entanglement is a foundational resource in quantum information science, underpinning applications across physics. However, detecting and quantifying entanglement remains a significant challenge. In this article, we introduce a variational quantum algorithm inspired by Uhlmann's theorem to quantify the Bures entanglement of general quantum states, a method that naturally extends to other quantum resources, including genuine multipartite entanglement, quantum discord, quantum coherence, and total correlations, while also enabling reconstruction of the closest free states. The algorithm requires a polynomial number of ancillary qubits and circuit depth relative to the system size, dimensionality, and free-state cardinality, making it scalable for practical implementations. Thus, it provides a versatile framework for quantifying quantum resources, demonstrated here through several applications.
Variational quantum algorithms have emerged as a powerful tool for harnessing the potential of near-term quantum devices to address complex challenges across quantum science and technology. Yet, the robust and scalable quantification of entanglement in many-body quantum systems remains a significant challenge, crucial for both advancing theoretical understanding and enabling practical applications. In this work, we propose a variational quantum algorithm to evaluate the M-partite geometric entanglement across arbitrary partitions of an N-qubit system into M parties. By constructing tailored variational ansatz circuits for both single- and multi-qubit parties, we optimize the overlap between a target quantum state and an M-partite variational separable state. This method provides a flexible and scalable approach for characterizing arbitrary M-partite entanglement in complex quantum systems of a given dimension. The accuracy of the proposed method is assessed by reproducing known analytical results. We further demonstrate its capability to evaluate entanglement among M parties for any given conventional or unconventional partitions of one- and two-dimensional spin systems, both near and at a quantum critical point. Our results establish the versatility of the variational approach in capturing different types of entanglement in various quantum systems, surpassing the capabilities of existing methods. Our approach offers a powerful methodology for advancing research in quantum information science, condensed matter physics, and quantum field theory. Additionally, we discuss its advantages, highlighting its adaptability to diverse system architectures in the context of near-term quantum devices.
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.
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.
Quantum entanglement is a fundamental resource in quantum information, but its characterization becomes increasingly challenging as the size of multipartite systems grows. This thesis investigates machine learning methods to detect and quantify entanglement in pure multiqubit states using only single qubit Pauli expectation values. Two complementary tasks are considered: the distinction between fully separable and entangled states, with particular attention to weakly entangled states close to the separability boundary, and the regression of normalized multipartite classical squashed entanglement. Several machine learning models are evaluated for detection, while multilayer perceptrons are employed for quantification. The low entanglement regime is investigated through dedicated datasets and targeted data augmentation strategies. The proposed approaches are studied for systems of 5, 10, 15, and 20 qubits and are further evaluated under realistic measurement conditions, including finite shot noise, asymmetric readout errors, calibration drift, and data acquired from real quantum processors. The study assesses the scalability, robustness, and practical limitations of data-driven entanglement characterization based on a linearly scaling set of local observables.
Multipartite entanglement is the premier resource for quantum technologies. Yet, its exact quantification in the laboratory is notoriously challenging, typically requiring the full knowledge of high dimensional quantum states. Here, we construct observable bounds to multipartite entanglement for systems of arbitrary size, which are functions of the local and global state purities, and correlation functions. First, we derive experimentally accessible upper and lower limits to both the bipartite entanglement of formation and the squashed entanglement of bipartite systems, by leveraging cornerstone results of quantum information theory: the entropy strong subadditivity inequality and the Koashi-Winter monogamy relation. Then, we convert them into bounds to the entanglement up to degree k for arbitrary states, and to the genuine k-partite entanglement, by employing a recently proposed method. Finally, we analytically and numerically test these results, by bounding the multipartite entanglement of several relevant states and mixtures, including the important classes of GHZ, Dicke, W states, and random pure states.
The design of detection strategies for multipartite entanglement stands as a central importance on our understanding of fundamental quantum mechanics and has had a substantial impact on quantum information applications. However, accurate and robust detection approaches are severely hindered, particularly when the number of nodes grows rapidly like in a quantum network. Here we present a general and operational framework that generates alternative entanglement witness for an arbitrary targeted state. The framework enjoys a systematic and high-efficient character and allows to substantiate genuine multipartite entanglement for a variety of states that arise naturally in practical situations and to dramatically outperform currently standard methods. With excellent noise tolerance, our framework should be broadly applicable to witness genuine multipartite entanglement in various practically scenarios and to facilitate the best use of entangled resources in the emerging area of quantum network.