Visualization of Three-Qubit Pure States with Separation of Local and Nonlocal Degrees of Freedom

TL;DR

This paper introduces a unified geometric visualization method for two- and three-qubit pure states that explicitly separates local properties from entanglement, aiding understanding and analysis of quantum states.

Contribution

It presents a novel, state-specific visualization framework that combines Bloch spheres and complex concurrences to clearly depict local and nonlocal quantum correlations.

Findings

Visualizes entanglement strength and phase simultaneously

Distinguishes states with identical entanglement but different interference

Provides intuitive insight into pairwise and tripartite entanglement

Abstract

Understanding the structure of multi-qubit quantum states is essential for both quantum information research and education, yet intuitive visualization beyond the single-qubit Bloch sphere remains challenging. In this work, we propose a unified geometric framework for visualizing two- and three-qubit pure states in which local degrees of freedom and entanglement degrees of freedom are explicitly separated. For two qubits, we combine Bloch-sphere representations of reduced density operators with a complex concurrence plotted on the complex plane, enabling simultaneous visualization of entanglement strength and phase structure. For three qubits, building on the generalized Schmidt decomposition, we introduce bipartite and GHZ-type tripartite complex concurrences, which, together with local Bloch vectors, provide a complete coordinate representation of the state. Unlike classification-based approaches, our method focuses on representing a given concrete state, revealing how local properties and nonlocal correlations coexist. The framework distinguishes states with identical entanglement magnitudes but different interference structures and provides intuitive insight into the balance between pairwise and genuinely tripartite entanglement. This approach offers both conceptual clarity and potential applications in quantum education and state analysis.