Bloch Sphere Representation

In summary: Bloch Sphere representation of qubits and how different vectors on the sphere can be represented as superpositions of basic states. It also clarified that the qubit will only remain static and unprecessing if it is not affected by external noise and the two states have the same energy. In summary, the Bloch Sphere represents qubits as vectors on a sphere, where any vector can be represented as a superposition of two basic states. The qubit will only remain static if it is not affected by external noise and the two states have the same energy.
  • #1
*FaerieLight*
43
0
There is something that I don't quite understand in relation to the Bloch Sphere representation of qubits. I've read that any vector on the sphere is a superposition of two basic states, like spin up and spin down, denoted by |1> and |0>.
So does this mean that if the vector is at z=0 (pointing towards the equator), then it has neither spin-up nor spin-down components?
If the vector is pointing up 45 degrees above the equator towards |1>, then how is this represented by equations as a superposition of |1> and |0> ?
Also, if a qubit is not affected by external noise, then does it remain a static unprecessing vector on the Bloch Sphere?
 
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  • #2
*FaerieLight* said:
So does this mean that if the vector is at z=0 (pointing towards the equator), then it has neither spin-up nor spin-down components?
No, it means that it is an equal superposition of spin-up and spin down,
$$
\frac{1}{\sqrt{2}} \left[ |0 \rangle + e^{i \phi} | 1 \rangle \right]
$$
*FaerieLight* said:
If the vector is pointing up 45 degrees above the equator towards |1>, then how is this represented by equations as a superposition of |1> and |0> ?
The general equation is
$$
\cos \left( \frac{\theta}{2} \right) |0 \rangle + \sin \left( \frac{\theta}{2} \right) e^{i \phi} | 1 \rangle
$$
so ##\theta = 135^\circ = 3\pi/4## results in
$$
\frac{\sqrt{2 - \sqrt{2}}}{2} |0 \rangle + \frac{\sqrt{2 + \sqrt{2}}}{2} e^{i \phi} | 1 \rangle
$$
so a superposition that is more ##|1 \rangle## than ##|0 \rangle##.

*FaerieLight* said:
Also, if a qubit is not affected by external noise, then does it remain a static unprecessing vector on the Bloch Sphere?
Only if the two states have the same energy. This is not true in most cases, so one will see a precession as the angle ##\phi## will be time-dependent.
 

1. What is the Bloch Sphere Representation?

The Bloch Sphere Representation is a visual representation of the state of a two-level quantum system, also known as a qubit. It was first introduced by physicist Felix Bloch in 1946 and has since become a popular tool for understanding and analyzing quantum systems.

2. How does the Bloch Sphere work?

The Bloch Sphere is a sphere with a point on its surface representing the state of the qubit. The north and south poles represent the two orthogonal basis states of the qubit, typically labeled as |0> and |1>. The equator of the sphere represents a superposition of these two states, with points on the equator representing different combinations of |0> and |1>. The rotation of the qubit state around the Bloch Sphere represents the evolution of the qubit under different quantum operations.

3. What is the significance of the Bloch Sphere Representation?

The Bloch Sphere Representation is a powerful tool for visualizing and understanding the behavior of qubits in quantum systems. It allows scientists to easily visualize the state of a qubit and track its evolution under different operations. It also helps in understanding concepts like quantum gates, measurements, and entanglement.

4. How is the Bloch Sphere related to other representations of qubits?

The Bloch Sphere is closely related to other representations of qubits, such as the Bloch vector and the density matrix. The Bloch vector is a three-dimensional vector that represents the state of a qubit on the surface of the Bloch Sphere. The density matrix is a mathematical tool that can be used to describe the state of a quantum system and can also be visualized on the Bloch Sphere.

5. What are the limitations of the Bloch Sphere Representation?

While the Bloch Sphere is a useful tool for visualizing qubit states, it has some limitations. It can only represent the state of a single qubit, and cannot be used to visualize multi-qubit systems. Additionally, the Bloch Sphere is limited to representing two-level quantum systems, so it cannot be used for higher-dimensional quantum systems. It is also not suitable for visualizing continuous variables, such as in quantum computing with continuous variables.

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