# Bloch sphere representation

Suppose you know that a qubit is either is in state $|+\rangle$ with probability $p$ or in state $|-\rangle$ with probability $1-p$. If this is the best you know about the qubit's state, where in the Bloch sphere would you represent this qubit?

• In your first sentence, are you specifying that the qubit is in a pure state? See response and comments below. – joshphysics Apr 5 '13 at 16:20

Since this is a homework question, here's a bit to get you started: I'll assume you mean that the qubit is in the pure state $$|\psi\rangle = \sqrt{p}|+\rangle + \sqrt{1-p}|-\rangle$$ The bloch sphere representation of pure states can be written as follows: $$|\psi\rangle = e^{i\phi}\sin(\theta/2)|+\rangle + \cos(\theta/2)|-\rangle$$ Post more of your work if you're still stuck and need guidance.
• That's not what the question says -- if I'm not mistaken, it says $\rho = p\vert+\rangle\langle+\vert + (1-p)\vert-\rangle\langle-\vert$. – Norbert Schuch Apr 5 '13 at 15:51
• @NorbertSchuch Normalization demands $\langle \psi|\psi\rangle = 1$. Given my expression for the state, one has $\langle \psi|\psi\rangle = \sqrt p^2 + \sqrt{1-p}^2 = p + 1-p=1$. – joshphysics Apr 5 '13 at 16:12