Bloch Sphere Representation

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?
 

DrClaude

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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,
$$
\frac{1}{\sqrt{2}} \left[ |0 \rangle + e^{i \phi} | 1 \rangle \right]
$$
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##.

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.
 

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