Bloch Sphere Approach for Two-Qubit Systems

• The Bloch sphere approach is a geometric model that represents quantum states as points on spheres, extending visualization from single-qubit pure states to complex two-qubit systems.
• It employs quaternion algebra and Hopf fibration to parameterize local and nonlocal properties, explicitly linking concurrence and phase relations in entangled states.
• This framework facilitates the analysis of quantum state evolution and two-qubit gate operations, offering clear insights into state decomposition and entanglement measurement.

The Bloch sphere approach is a geometrical formalism that represents quantum states and their evolution by mapping the state vectors of quantum systems—especially two-level (qubit) or composite systems—onto high-dimensional spheres or related geometric structures. It extends the traditional visualization of single-qubit pure states as points on the surface of a unit 2-sphere (the Bloch sphere) to higher-dimensional spaces and multipartite entanglement. For two-qubit pure states, the approach encapsulates both local properties and nonlocal correlations by parameterizing the state through three unit 2-spheres (Bloch and entanglement spheres) and a global phase, linking these geometric variables via Hopf fibration and quaternionic relations. This geometrically unified perspective provides operational insights into both quantum state evolution and gate functionality, offering an explicit framework for the decomposition and visualization of entanglement, concurrence, and local phases.

1. Geometric Construction: From Single-Qubit to Two-Qubit Bloch Sphere

The conventional Bloch sphere maps a qubit's pure state $$|\psi\rangle = \cos(\theta/2)|0\rangle + e^{i\varphi}\sin(\theta/2)|1\rangle$$ to a unique point on the 2-sphere $S^2$, with the polar angle $\theta$ and azimuthal angle $\varphi$ corresponding to latitude and longitude (Wie, 2014). This provides a complete correspondence between Hilbert space rays and points on $S^2$, with an additional $U(1)$ phase factor being physically irrelevant for pure state kinematics of a single qubit.

For two-qubit pure states, $$|\Psi\rangle = a|00\rangle + B|01\rangle + y|10\rangle + 8|11\rangle$$, the approach parameterizes the general state using three unit 2-spheres and an overall phase, explicitly:

• The coordinates $(\Theta_A, \phi_A)$ for qubit-A Bloch sphere (local state of qubit A),
• The coordinates $(\Theta_B, \phi_B)$ for qubit-B Bloch sphere (local state of qubit B),
• The pair $(x, \varphi)$ for the entanglement sphere (degree and phase of concurrence),
• An additional global phase factor, rendered as a Hopf fiber (S¹).

The full parameter space becomes $S^2 \times S^2 \times S^2 \times S^1$ (or, equivalently, $S^7$, since a two-qubit pure state is a point on the unit 7-sphere in $\mathbb{C}^4$ up to global phase).

For separable states, the model reduces such that the two Bloch spheres parameterize the tensor product structure, and the entanglement sphere is redundant. For entangled states, the third sphere (the "entanglement sphere") encodes nonlocal features—specifically, the concurrence and its nonlocal relative phase (Wie, 2014).

2. Explicit Parameterization via Quaternionic Formalism and Hopf Fibration

The mapping from state amplitudes to geometric variables uses quaternionic notation, providing a compact algebraic framework that naturally incorporates noncommutativity and rotations. The four amplitudes are assembled into two quaternions:

• $q_0 = a + B\,j$, $q_1 = y + 8\,j$, with $i,j,k$ the quaternion units.

A representative parameterization is of the form:

$$\Psi = (a + B\,j) = \cos\Theta_A\,e^{i\phi_A} + \sin\Theta_A\,e^{i\phi_A}\,t$$

where $t$ is a pure unit quaternion specifying the entanglement, given by:

$$t = i\sin x\cos\varphi + j\sin x\sin\varphi + k\cos x$$

with $t^2 = -1$. This provides the geometric structure for two (quasi-)Bloch spheres and the entanglement sphere, while the additional phase factor from the Hopf fibration accounts for the global phase degree of freedom (Wie, 2014).

The Hopf fibration acts as the key geometric underpinning: the $S^7$ state space is decomposed into an $S^4$ base space (itself further reducible to two $S^2$ spheres) and an $S^3$ fiber, with the S³ structure corresponding to local SU(2) rotations of a qubit. This mapping ensures geometric consistency under local unitary transformations and links the kinematics of physical parameters directly to quantum gate actions.

3. The Entanglement Sphere: Parameterization of Nonlocality

The introduction of the entanglement sphere distinguishes the geometric representation of two-qubit states from mere products of single-qubit Bloch spheres. Its spherical coordinates $(x, \varphi)$ serve two fundamental roles:

• The concurrence $c$, a measure of entanglement, is given algebraically as $c = \sin \Theta_A \sin \phi_A \sin x$. Thus, $c=0$ (for $x=0$ or $x = \pi$) corresponds to a separable state, and $c=1$ (for $x = \pi/2$) to maximal entanglement.
• The azimuthal angle $\varphi$ specifies a nonlocal relative phase that cannot be adjusted by local operations; it is not removable by unitaries acting solely on one qubit.

This nonlocal relative phase manifests in physical phenomena such as interference and in the operation of multi-qubit gates that are sensitive to the global entanglement topology of the state. The entanglement sphere thus provides a geometric quantification of "how far" a state is from being separable, and specifies the nonlocal phase characteristics fully (Wie, 2014).

4. Geometry of Two-Qubit Gates and State Evolution

The full representation accommodates not only static visualization but also dynamic trajectories under two-qubit unitary operations. In the parameter space constructed (three S² spheres and a phase), two-qubit unitaries, including canonical gates such as CNOT, CZ, and SWAP, correspond to paths on the composite sphere, with local (single-qubit) rotations mapped to movements on the respective Bloch spheres and nonlocal gate-induced entangling actions traced as shifts in the entanglement sphere coordinates.

The Hopf fibration guarantees that global phase and local equivalence classes are consistently tracked and that physically equivalent states are mapped to the same geometric locus modulo the fiber action. The geometric clarity of such representations is particularly useful for tracking State evolution, quantum circuit design, and interpretation of gate-mixedness and entanglement manipulation (Wie, 2014).

5. Mathematical Relations and Unification with Quaternions

A single quaternionic relation encompasses the mapping from analytic amplitudes to geometric variables:

$$a + B\,j - y + 8\,j = (\text{expression involving}\ \cos\Theta_A, \sin\Theta_A, t, \text{and phases})$$

This compactly encapsulates all seven required real parameters for specifying a two-qubit pure state on $S^7$, unifying the local Bloch sphere variables with nonlocal entanglement sphere data. The use of quaternions is essential since two-qubit operations inherently involve noncommuting rotations and spinorial transformations in the combined space.

The non-commutative character of quaternions ensures faithful representation of quantum rotations and entangling operations, as conventional complex-number algebra is insufficient for capturing all required degrees of freedom and phase phenomena in composite systems (Wie, 2014).

6. Implications and Applications in Quantum Information

This geometrically explicit extension of the Bloch sphere model provides:

• Visualization frameworks for both local and global quantum information (enabling visualization of entanglement generation, gate-induced state transfer, and relative phase manipulation).
• Gate analysis tools, permitting the representation and design of two-qubit unitary gates in terms of simple geometric operations on spheres, and facilitating analytic calculation of their effect on both local and nonlocal parameters.
• Parameter extraction techniques, offering explicit equations for extracting local phases and concurrence directly from amplitudes, with applications in quantum state tomography and error correction.
• Foundational insights, as the model's use of Hopf fibration and non-commutative algebra exposes deep connections between quantum kinematics, fiber bundle theory, and topological features of quantum state space.

These consequences support both practical quantum information engineering and further theoretical exploration of quantum nonlocality. The scheme naturally generalizes to higher-order systems by analogy, although the complexity of the corresponding fiber structures and parameterizations increases.

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In summary, the Bloch sphere approach for two-qubit pure states extends the single-qubit geometric representation to a composite structure of three 2-spheres and a phase factor, intrinsically linking local qubit properties and nonlocal entanglement. This construction leverages quaternion algebra and Hopf fibration to deliver a unified geometric and operational framework for the state space and dynamics of two-qubit systems, enabling precise exploration and manipulation of both the local and global features of quantum states (Wie, 2014).