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

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.

Figures (2)

• Figure 1: Examples of two-qubit state visualization in the proposed framework. (a) A general two-qubit state. The left panel shows the reduced density operators $\rho_1$ and $\rho_2$ in Bloch sphere, while the right panel shows the complex concurrence $\tilde{C}$ on the complex plane. (b) Visualization of $(\ket{00} + \mathrm{e}^{i\alpha} \ket{11})/\sqrt{2}$. Both reduced density operators are located at the origin (maximally mixed states). The complex concurrence lies on the unit circle at $\mathrm{e}^{i\alpha}$. (c) Example of a product state. The local states are pure and thus lie on the surface of the Bloch sphere. Since the state is separable, the complex concurrence is zero.
• Figure 2: Examples of three-qubit state visualization in the proposed framework. (a) A general three-qubit state. The left panel shows three reduced density operators $\rho_1, \rho_2, \rho_3$ on Bloch spheres, and the right panel shows the four complex concurrences $\tilde{C}_{12}, \tilde{C}_{13}, \tilde{C}_{23}, \tilde{C}_{123}$ on the complex plane. (b) Visualization of $(\ket{000} + \mathrm{e}^{i\alpha} \ket{111})/\sqrt{2}$. All reduced density operators lie at the origin (maximally mixed states). Only the GHZ-type complex concurrence $\tilde{C}_{123}$ lies on the unit circle at $\mathrm{e}^{i\alpha}$, while the others vanish. (c) Visualization of $(\ket{000}+\ket{101}+\ket{110})/\sqrt{3}$. The GHZ-type complex concurrence $\tilde{C}_{123}$ is zero, and the bipartite complex concurrences satisfy $\tilde{C}_{12}=\tilde{C}_{13}=\tilde{C}_{23}=2/3$. (d) Example of a product state. Since the local states are pure, the points lie on the surfaces of the Bloch spheres. As the state is separable, all complex concurrences shown in the right panel are zero.