Quantum entanglement, a cornerstone of quantum mechanics, remains challenging to classify, particularly in multipartite systems. Here, we present a new interpretation of entanglement classification by revealing a profound connection to Galois groups, the algebraic structures governing polynomial symmetries. This approach not only uncovers hidden geometric relationships between entangled quantum states and polynomial roots but also introduces a method for quantifying entanglement in multi-qubit symmetric states. By reframing the classification of GHZ, W, and separable states within the structure of Galois symmetries, we establish a previously unrecognized hierarchy in their entanglement properties. This work bridges the mathematical elegance of Galois theory with the complexities of quantum mechanics, opening pathways for advances in quantum computing and information theory.
Quantum entanglement, a cornerstone of quantum mechanics, remains challenging to classify, particularly in multipartite systems. Here, we present a new interpretation of entanglement classification by revealing a profound connection to Galois groups, the algebraic structures governing polynomial symmetries. This approach not only uncovers hidden geometric relationships between entangled quantum states and polynomial roots but also introduces a method for quantifying entanglement in multi-qubit symmetric states. By reframing the classification of GHZ, W, and separable states within the structure of Galois symmetries, we establish a previously unrecognized hierarchy in their entanglement properties. This work bridges the mathematical elegance of Galois theory with the complexities of quantum mechanics, opening pathways for advances in quantum computing and information theory.
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.
One of the key manifestations of quantum mechanics is the phenomenon of quantum entanglement. While the entanglement of bipartite systems is already well understood, our knowledge of entanglement in multipartite systems is still limited. This dissertation covers various aspects of the quantification of entanglement in multipartite states and the role of symmetry in such systems. Firstly, we establish a connection between the classification of multipartite entanglement and knot theory and investigate the family of states that are resistant to particle loss. Furthermore, we construct several examples of such states using the Majorana representation as well as some combinatorial methods. Secondly, we introduce classes of highly-symmetric but not fully-symmetric states and investigate their entanglement properties. Thirdly, we study the well-established class of Absolutely Maximally Entangled (AME) quantum states. On one hand, we provide construction of new states belonging to this family, for instance, an AME state of 4 subsystems with six levels each, on the other, we tackle the problem of equivalence of such states. Finally, we present a novel approach for the general problem of verification of the equivalence between any pair of arbitrary quantum states based on a single polynomial entanglement measure.
Quantum mechanics could be studied through polynomial algebra, as has been demonstrated by a work (“An approach to first-principles electronic structure calculation by symbolic-numeric computation” by A. Kikuchi). We carry forward the algebraic method of quantum mechanics through algebraic number theory; the basic equations are represented by the multivariate polynomial ideals; the symbolic computations process the ideal and disentangle the eigenstates as the algebraic variety; upon which one canbuild the Galois extension of the number field, in analogy with the univariate polynomial case, to investigate the hierarchy of solutions; the Galois extension is accompanied with the group operations, which permute the eigenstates from one to another, and furnish the quantum system with a non-apparent symmetry. Besides, this sort of algebraic quantum mechanics is an embodiment of the class field theory; some of the important consequences of the latter emerge in quantum mechanics. We shall demonstrate these points through simple models; we will see the use of computational algebra facilitates such sort of analysis, which might often be complicated if we try to solve them manually.
Abstract We approach multipartite entanglement classification in the symmetric subspace in terms of algebraic geometry, its natural language. We show that the class of symmetric separable states has the structure of a Veronese variety and that its k -secant varieties are SLOCC invariants. Thus SLOCC classes gather naturally into families. This classification presents useful properties such as a linear growth of the number of families with the number of particles, and nesting, i.e. upward consistency of the classification. We attach physical meaning to this classification through the required interaction length of parent Hamiltonians. We show that the states W N and GHZ N are in the same secant family and that, effectively, the former can be obtained in a limit from the latter. This limit is understood in terms of tangents, leading to a refinement of the previous families. We compute explicitly the classification of symmetric states with N ⩽ 4 qubits in terms of both secant families and its refinement using tangents. This paves the way to further use of projective varieties in algebraic geometry to solve open problems in entanglement theory.
This paper introduces the Entanglement Dimension, a novel group invariant derived from the application of quantum information theory concepts to algebraic structures. Traditional group invariants, such as order, nilpotency class, and solvability length, offer crucial insights but often fail to distinguish between non-isomorphic groups sharing many common properties. By mapping group elements or their representations to quantum states and subsequently quantifying the entanglement present in superpositions formed from these states, we propose a new measure that captures nuanced aspects of group structure. We rigorously define the Entanglement Dimension and demonstrate its computation for various finite groups, including cyclic, dihedral, and symmetric groups. Our results indicate that this invariant can differentiate between groups previously indistinguishable by classical methods, providing a finer characterization of their internal symmetries and interactions. The Entanglement Dimension offers a fresh perspective on group theory, bridging algebraic structures with quantum mechanical principles and opening new avenues for research in both fields, particularly in areas like quantum computing and fundamental physics.
Quantum entanglement, a cornerstone of quantum mechanics, presents a unique challenge to computational efficiency. Traditional methods often struggle to efficiently handle the intricate calculations involved in quantum systems. This paper proposes a novel approach utilizing algebraic vector representation to transform entanglement calculations into a set of algebraic operations, aiming to dramatically improve computational speed and provide deeper insights into the fundamental nature of quantum entanglement. We establish a new method for representing entangled states as algebraic vectors, leveraging the geometric properties of string theory to achieve this transformation. The core mechanism involves the development of a specialized algebraic vector representation that allows for efficient computation through a sequence of transformations, ultimately revealing the underlying mathematical structure of entangled states. This research explores the potential for quantum entanglement computation to be revolutionized by this algorithmic shift.