Bloch Sphere Representation

Table of Contents

1. Introduction
2. What Is the Bloch Sphere?
3. State of a Qubit
4. Parametrization of Qubit States
5. General Form of a Single-Qubit Pure State
6. Normalization and Global Phase
7. Coordinate Mapping on the Sphere
8. Geometrical Structure of the Bloch Sphere
9. Visualization of Quantum States
10. Basis States on the Bloch Sphere
11. Superposition and Equatorial States
12. Phase Information on the Bloch Sphere
13. Action of Quantum Gates on Bloch Sphere
14. Pauli Gates as Rotations
15. Hadamard Gate and the X-Z Plane
16. Phase Gates and Z-Axis Rotations
17. Bloch Sphere and Mixed States
18. The Bloch Vector and Density Matrix
19. Radius of the Bloch Vector
20. Quantum Operations as Rotations
21. Measurement and Collapse on the Bloch Sphere
22. Entanglement and the Limitation of Bloch Sphere
23. Advantages and Limitations of the Bloch Sphere
24. Applications in Quantum Computing
25. Conclusion

1. Introduction

The Bloch sphere is one of the most intuitive and widely used visualizations in quantum computing. It provides a geometric representation of the state of a single qubit and offers deep insights into quantum operations, measurement, and superposition.

2. What Is the Bloch Sphere?

The Bloch sphere is a unit sphere in \( \mathbb{R}^3 \) used to represent pure states of a single qubit. Each point on the surface corresponds to a possible pure quantum state.

3. State of a Qubit

A qubit can be written in the general form:

\[
|\psi\rangle = \alpha|0\rangle + \beta|1\rangle
\]

where \( \alpha, \beta \in \mathbb{C} \), and \( |\alpha|^2 + |\beta|^2 = 1 \)

4. Parametrization of Qubit States

Any pure state of a qubit can be expressed as:

\[
|\psi\rangle = \cos\left(\frac{\theta}{2}\right)|0\rangle + e^{i\phi} \sin\left(\frac{\theta}{2}\right)|1\rangle
\]

with \( 0 \leq \theta \leq \pi \), \( 0 \leq \phi < 2\pi \)

5. General Form of a Single-Qubit Pure State

This parametrization maps quantum states to points on a unit sphere:

• \( \theta \): polar angle from the Z-axis
• \( \phi \): azimuthal angle in the X-Y plane

6. Normalization and Global Phase

The global phase \( e^{i\gamma} \) is physically irrelevant. Hence, every pure state is represented by a point on the surface of the Bloch sphere, not inside.

7. Coordinate Mapping on the Sphere

The state vector corresponds to a point:

\[
\vec{r} = (\sin\theta \cos\phi, \sin\theta \sin\phi, \cos\theta)
\]

This vector \( \vec{r} \) is called the Bloch vector.

8. Geometrical Structure of the Bloch Sphere

• The North pole \( (\theta = 0) \): \( |0\rangle \)
• The South pole \( (\theta = \pi) \): \( |1\rangle \)
• The Equator \( (\theta = \pi/2) \): superpositions like \( (|0\rangle + e^{i\phi}|1\rangle)/\sqrt{2} \)

9. Visualization of Quantum States

Each qubit state corresponds to a point on the surface of the Bloch sphere, enabling:

• Visualization of superposition
• Representation of gate operations as rotations

10. Basis States on the Bloch Sphere

• \( |0\rangle \): (0, 0, 1)
• \( |1\rangle \): (0, 0, -1)
• \( (|0\rangle + |1\rangle)/\sqrt{2} \): (1, 0, 0)
• \( (|0\rangle – |1\rangle)/\sqrt{2} \): (-1, 0, 0)

11. Superposition and Equatorial States

The equator contains all equal superpositions of \( |0\rangle \) and \( |1\rangle \) with varying relative phases:

\[
\frac{1}{\sqrt{2}}(|0\rangle + e^{i\phi}|1\rangle)
\]

These lie on the X-Y plane.

12. Phase Information on the Bloch Sphere

The azimuthal angle \( \phi \) encodes the relative phase between the basis states. This phase plays a crucial role in interference and algorithm performance.

13. Action of Quantum Gates on Bloch Sphere

Unitary operations correspond to rotations of the Bloch vector. For example:

• X gate: \( \pi \)-rotation around X-axis
• Y gate: \( \pi \)-rotation around Y-axis
• Z gate: \( \pi \)-rotation around Z-axis

14. Pauli Gates as Rotations

\[
X = R_x(\pi), \quad Y = R_y(\pi), \quad Z = R_z(\pi)
\]

Each Pauli gate corresponds to a 180° rotation around its respective axis.

15. Hadamard Gate and the X-Z Plane

Hadamard maps \( |0\rangle \to (|0\rangle + |1\rangle)/\sqrt{2} \), placing it on the X-axis. It corresponds to a rotation that brings poles to equator and vice versa.

16. Phase Gates and Z-Axis Rotations

• S gate: \( \pi/2 \) rotation around Z
• T gate: \( \pi/4 \) rotation around Z

These gates change the phase, affecting interference.

17. Bloch Sphere and Mixed States

Mixed states are represented as points inside the Bloch sphere (not on the surface). The density matrix \( \rho \) maps to a Bloch vector:

\[
\rho = \frac{1}{2}(\mathbb{I} + \vec{r} \cdot \vec{\sigma})
\]

18. The Bloch Vector and Density Matrix

For a mixed state, the vector \( \vec{r} \) satisfies \( |\vec{r}| < 1 \). Pure states have \( |\vec{r}| = 1 \).

19. Radius of the Bloch Vector

The radius gives a measure of purity:

• \( r = 1 \): pure state
• \( r < 1 \): mixed state
• \( r = 0 \): maximally mixed (totally random)

20. Quantum Operations as Rotations

Quantum operations (unitary gates) act as rotations of the Bloch vector. Decoherence and noise shrink the Bloch vector toward the center.

21. Measurement and Collapse on the Bloch Sphere

Measurement along Z-axis:

• Projects state to either North or South pole
• Collapses vector to \( |0\rangle \) or \( |1\rangle \)

22. Entanglement and the Limitation of Bloch Sphere

The Bloch sphere applies only to single qubits. It does not capture multi-qubit correlations or entanglement — for those, other tools are needed (like tensor networks).

23. Advantages and Limitations of the Bloch Sphere

Advantages:

• Intuitive visualization
• Insight into gate actions and interference

Limitations:

• Cannot represent entangled states
• Only valid for 2D (single-qubit) Hilbert spaces

24. Applications in Quantum Computing

• Visualizing gate effects
• Understanding superposition and phase
• Educational and debugging purposes

Quantum software platforms often provide Bloch visualizations (e.g., Qiskit, QuTiP).

25. Conclusion

The Bloch sphere provides an elegant and powerful way to visualize qubit states, their evolution, and how quantum gates act. It is a cornerstone of quantum computing education and continues to aid in intuitive understanding of core quantum phenomena.