# I How to understand the Bloch sphere in the quantum computation?

#### Haorong Wu

I've read that $\left | \psi \right > =cos \frac \theta 2 \left | 0 \right > + e^{i \phi} sin \frac \theta 2 \left | 1 \right >$, and the corresponding point in the Bloch sphere is as the fig below shows.

I think $\left | 0 \right >$ and $\left | 1 \right >$ are orthonormal vectors, then why they seem to apear parallel in the Bloch sphere?

Also, I can understand the $cos \frac \theta 2 \left | 0 \right >$ part, but I cannot understand how $e^{i \phi} sin \frac \theta 2 \left | 1 \right >$ part can match the fig.

Thanks!

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#### DrClaude

Mentor
Consider a spin-1/2 particle. The two eigenvalues of $S_z$ are $\pm \hbar/2$, which correspond to opposite points along $z$. The fact that $|+z\rangle$ and $|-z\rangle$ are orthogonal states must not be confused with the orthogonality of the cartesian axes.

Likewise, consider that
$$|\pm x \rangle = \frac{1}{\sqrt{2}} \left( |+z \rangle \pm |-z \rangle \right) \\ |\pm y \rangle = \frac{1}{\sqrt{2}} \left( |+z \rangle \pm i |-z \rangle \right)$$
This is exactly what is transposed to the Bloch sphere.

#### cosmik debris

I think $\left | 0 \right >$ and $\left | 1 \right >$ are orthonormal vectors, then why they seem to apear parallel in the Bloch sphere?
Spin half particles take 720 degrees to return to their original state.

Cheers

"How to understand the Bloch sphere in the quantum computation?"

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