# How to plot a qubit on the Bloch sphere?

I've been reading pages such as this one: http://comp.uark.edu/~jgeabana/blochapps/bloch.html

Which talk about the Bloch sphere, but I've been unable to figure out how to plot states on the sphere due to both states being a complex number, thus resulting in 4 "coordinates".

Many sources say it's only the relative angle between the states that matters, but that seems odd to me, since there is an absolute $|0\rangle$ and $|1\rangle$ state, which makes it think that both imaginary parts of the complex values matter.

So, how would I plot things like these, or at least, how would I convert them to spherical coordinates for plotting on the bloch sphere?

$$1/\sqrt{2}(|0\rangle + |1\rangle)$$ $$1/\sqrt{2}(|0\rangle - |1\rangle)$$ $$1/2(\sqrt{3}|0\rangle + |1\rangle)$$

• I dare you to pass the turing test Normal Human! Your response was too quick :P Oct 16, 2015 at 20:12

$$\cos\theta/2 = 1/\sqrt 2, \quad e^{i\phi}\sin\theta/2 = 1/\sqrt 2$$

Hence--given the range of values of $\theta$ and $\phi$--we have $\theta = \pi/2$ and $\phi = 0$.

Proceed similarly for the other qubits.

An important thing to remember about qubits is that we don't care about overall phase. Hence if we have a qubit of the form

$$|\psi\rangle = e^{i\alpha}a|0\rangle + e^{i\beta}b|1\rangle = e^{i\alpha} \left( a|0\rangle + e^{i(\beta-\alpha)}b|1\rangle \right)$$

we find the Bloch representation of $a|0\rangle + e^{i(\beta-\alpha)}b|1\rangle$.

• Thanks! Why is it that we don't care about overall phase? if the answer isn't too long for a comment hehe Oct 16, 2015 at 20:40
• Because a multiple of a phase $e^{i\theta}$ of a coefficient doesn't affect the probability of being in a given state. It is however worth keeping track of the phase difference between the components so we don't dispense with phase altogether. Oct 16, 2015 at 20:44
• oh got it. thanks! Oct 16, 2015 at 20:44
• @Alan Wolfe If you wanted a striclty formal explanation, here is one. Jan 7, 2020 at 17:38
• Good answer! I like it! Apr 7 at 4:12