Entanglement or separability: The choice of how to factorize the algebra of a density matrix

TL;DR

This paper explores how the perception of entanglement or separability in quantum states depends on the factorization of the algebra, highlighting the implications for pure and mixed states, with detailed analysis of specific quantum states.

Contribution

It introduces a framework for understanding how different factorizations affect entanglement, extending the interpretation of quantum states and their applications.

Findings

Pure states can switch between entangled and separable via unitary transformations.

Mixed states require a minimal amount of mixedness to change entanglement status.

Geometric features of specific states like GHZ, Werner, and Gisin states are analyzed.

Abstract

We discuss the concept of how entanglement changes with respect to different factorizations of the total algebra which describes the quantum states. Depending on the considered factorization a quantum state appears either entangled or separable. For pure states we always can switch unitarily between separability and entanglement, however, for mixed states a minimal amount of mixedness is needed. We discuss our general statements in detail for the familiar case of qubits, the GHZ states, Werner states and Gisin states, emphasizing their geometric features. As theorists we use and play with this free choice of factorization, which is naturally fixed for an experimentalist. For theorists it offers an extension of the interpretations and is adequate to generalizations, as we point out in the examples of quantum teleportation and entanglement swapping.