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arXiv:2003.01699 · Physics

Two-Qubit Bloch Sphere

Chu-Ryang Wie

Published Aug 2, 2020• 32 citations
arXivHTMLPDF

Abstract

Three unit spheres were used to represent the two-qubit pure states. The three spheres are named the base sphere, entanglement sphere, and fiber sphere. The base sphere and entanglement sphere represent the reduced density matrix of the base qubit and the non-local entanglement measure, concurrence, while the fiber sphere represents the fiber qubit via a simple rotation under a local single-qubit unitary operation; however, in an entangled bipartite state, the fiber sphere has no information on the reduced density matrix of the fiber qubit. When the bipartite state becomes separable, the base and fiber spheres seamlessly become the single-qubit Bloch spheres of each qubit. Since either qubit can be chosen as the base qubit, two alternative sets of these three spheres are available, where each set fully represents the bipartite pure state, and each set has information of the reduced density matrix of its base qubit. Comparing this model to the two Bloch balls representing the reduced density matrices of the two qubits, each Bloch ball corresponds to two unit spheres in our model, namely, the base and entanglement spheres. The concurrence–coherence complementarity is explicitly shown on the entanglement sphere via a single angle.

DOI
10.3390/physics2030021

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Three unit spheres were used to represent the two-qubit pure states. The three spheres are named the base sphere, entanglement sphere, and fiber sphere. The base sphere and entanglement sphere represent the reduced density matrix of the base qubit and the non-local entanglement measure, concurrence, while the fiber sphere represents the fiber qubit via a simple rotation under a local single-qubit unitary operation; however, in an entangled bipartite state, the fiber sphere has no information on the reduced density matrix of the fiber qubit. When the bipartite state becomes separable, the base and fiber spheres seamlessly become the single-qubit Bloch spheres of each qubit. Since either qubit can be chosen as the base qubit, two alternative sets of these three spheres are available, where each set fully represents the bipartite pure state, and each set has information of the reduced density matrix of its base qubit. Comparing this model to the two Bloch balls representing the reduced density matrices of the two qubits, each Bloch ball corresponds to two unit spheres in our model, namely, the base and entanglement spheres. The concurrence–coherence complementarity is explicitly shown on the entanglement sphere via a single angle.
We represent a two-qubit density matrix in the basis of Pauli matrix tensor products, with the coefficients constituting a Bloch matrix, analogous to the single qubit Bloch vector. We find the quantum state positivity requirements on the Bloch matrix components, leading to three important inequalities, allowing us to parametrize and visualize the two-qubit state space. Applying the singular value decomposition naturally separates the degrees of freedom to local and nonlocal, and simplifies the positivity inequalities. It also allows us to geometrically represent a state as two entangled Bloch spheres with superimposed correlation axes. It is shown that unitary transformations, local or nonlocal, have simple interpretations as axis rotations or mixing of certain degrees of freedom. The nonlocal unitary invariants of the state are then derived in terms of local unitary invariants. The positive partial transpose criterion for entanglement is generalized, and interpreted as a reflection, or a change of a single sign. The formalism is used to characterize maximally entangled states, and generalize two qubit isotropic and Werner states.
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.
We extend Bloch sphere formalism to pure two-qubit systems. Combining insights from Geometric Algebra and the analysis of entanglement in different conjugate bases we identify two Bloch sphere geometry that is suitable for representing maximally entangled states. It turns out that the relative direction of the coordinate axes of the two Bloch spheres may be used to describe the states. Moreover, the coordinate axes of one Bloch sphere should be rignt-handed and those of the other one should be left-handed. We describe and depict separable and maximally entangled states as well as entangling and non-entangling rotations. We also offer a graphical representation of the workings of a CNOT gate for different inputs. Finally, we provide a way to also represent partially entangled states and describe entanglement measures related to the surface area of the sphere enclosing the state representation.
The Bloch sphere is an elegant tool for representing single-qubit states. However, a widely accepted generalization for multi-qubit systems with entanglement remains absent. We propose a novel geometric model extending the Bloch sphere representation to arbitrary NNN-qubit systems using 2N−12^N-12N−1 spheres. We demonstrate that any pure 2-qubit state is uniquely described by three spheres: two for individual qubits and a third encapsulating bipartite entanglement. Generalizing this, we establish an NNN-qubit parameterization through the hierarchical application of controlled rotation gates along the ZZZ and YYY axes. We formally prove a strict bijection between the standard state vector representation and our model's angular parameters. This framework provides an intuitive visualization of multiple entanglement, offering potential computational advantages for quantum simulators and new analytical perspectives on quantum gates.
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.
I propose a simple visualization for pure two-qubit states, constructed from three components: two Bloch balls for the marginal qubits (the mixed states obtained by ignoring the other qubit, respectively) and an entanglement component consisting of a rotoreflection and a contraction. I prove that this picture accurately captures the local and global structure of the relevant state space without gaps or degeneracies: the underlying parametrization is a global diffeomorphism onto the state space. Finally, I use this picture to visualize conditional states, the entanglement degree(s) of freedom, and single qubit rotations.