Galois Symmetries in the Classification and Quantification of Quantum Entanglement

TL;DR

This paper introduces a novel approach to classifying and quantifying quantum entanglement by leveraging Galois group symmetries, revealing new geometric and hierarchical insights into multipartite quantum states.

Contribution

It establishes a new connection between Galois theory and quantum entanglement classification, providing a framework for understanding entanglement structures in multi-qubit systems.

Findings

Revealed geometric relationships between entangled states and polynomial roots.

Uncovered a hierarchy in entanglement properties of GHZ, W, and separable states.

Proposed a method for quantifying entanglement using Galois symmetries.

Abstract

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.