Entanglement in Quantum Process Algebra

Samuel L. Braunstein

Jincheng Zhang

This paper explores the application of non-associative algebra to the representation and manipulation of quantum state systems. The current mathematical landscape often falls short in adequately capturing the intricate and non-linear entanglement patterns inherent in quantum mechanics. We propose a novel algebraic structure, termed "non-associative entanglement," designed to address these limitations by providing a framework for representing and calculating entanglement with unprecedented precision. The core claim is to establish a new class of non-associative algebras capable of accurately modeling and processing complex quantum states, facilitating automated calculations of entanglement and superposition. This work will detail the mathematical foundations of this approach, showcasing its potential to revolutionize quantum state representation and computation.

Vladimir V. Kornyak

Abstract In this paper, we investigate the behavior of quantum entanglement in the process of unitary evolution in constructive models of multicomponent quantum systems. Symmetry groups of quantum systems that admit the occurrence of geometric structures associated with quantum entanglement are described. Algorithms for dynamic simulation of quantum entanglement are based on methods of computer algebra and computational group theory. Some examples of practical computations are presented.

Shengjun Wu

Timothy F. Havel, Chris Doran

This paper develops a geometric model for coupled two-state quantum systems (qubits), which is formulated using geometric (aka Clifford) algebra. It begins by showing how Euclidean spinors can be interpreted as entities in the geometric algebra of a Euclidean vector space. This algebra is then lifted to Minkowski space-time and its associated geometric algebra, and the insights this provides into how density operators and entanglement behave under Lorentz transformations are discussed. The direct sum of multiple copies of space-time induces a tensor product structure on the associated algebra, in which a suitable quotient is isomorphic to the matrix algebra conventionally used in multi-qubit quantum mechanics. Finally, the utility of geometric algebra in understanding both unitary and nonunitary quantum operations is demonstrated on several examples of interest in quantum information processing.