Entanglement beyond tensor product structure: algebraic aspects of quantum non - separability

TL;DR

This paper explores quantum non-separability using an algebraic approach that does not rely on tensor product structures, providing new insights into entanglement's relativity and a measure based on total correlation.

Contribution

It introduces an algebraic formalism for quantum entanglement that emphasizes observable-based partitions, offering a novel way to characterize and quantify entanglement.

Findings

Algebraic approach clarifies the relativity of entanglement to measured observables.

Pure state non-separability is characterized by total correlation.

Correlation matrix norm serves as a universal measure of non-separability.

Abstract

Algebraic approach to quantum non - separability is applied to the case of two qubits. It is based on the partition of the algebra of observables into independent subalgebras and the tensor product structure of the Hilbert space is not exploited. Even in this simple case, such general formulation has some advantages. Using algebraic formalism we can explicitly show the relativity of the notion of entanglement to the observables measured in the system and characterize separable and non - separable pure states. As a universal measure of non - separability of pure states we propose to take so called total correlation. This quantity depends on the state as well as on the algebraic partition. Its numerical value is given by the norm of the corresponding correlation matrix