I have sketched out some proofs for these things, but can you offer any help?

1) Let $\displaystyle |\phi\rangle = \lambda|0\rangle + \mu|1\rangle$ be a unit vector in a hilbert space, $\displaystyle H$. So $\displaystyle |\lambda|^2 + |\mu|^2 = 1$ and $\displaystyle \lambda = r_1e^{i\theta_1}, \: \mu = r_2e^{i\theta_2}$. Show that we can rotate $\displaystyle |\phi\rangle$ so that the coefficient of $\displaystyle |0\rangle$ is real and nonnegative.

My answer:

Now take $\displaystyle \lambda = r_1e^{i\theta_1}, \: \mu = r_2e^{i\theta_2}$. Then $\displaystyle |\phi\rangle = r_1e^{i\theta_1}|0\rangle + r_2e^{i\theta_2}|1\rangle$. Now rotate it through by $\displaystyle e^{-i\theta_1}$.

$\displaystyle |\phi\rangle = r_1e^{i\theta_1}e^{-i\theta_1}|0\rangle + r_2e^{i\theta_2}e^{-i\theta_1}|1\rangle$

$\displaystyle = r_1e^0|0\rangle + r_2e^{i\theta_2-i\theta_1}|1\rangle$

$\displaystyle = r|0\rangle + r_2e^{i(\theta_2-\theta_1)}|1\rangle$.

Hence the coefficient of $|0\rangle$ is real and nonnegative.

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2) And let $\displaystyle |\omega\rangle = \lambda'|0\rangle + \mu'|1\rangle; \: \lambda',\mu' \in \mathbb{R}_{\ge 0}$. Show that each such vector in $\displaystyle H$ corresponds to a unique point on the Bloch sphere.

My answer:

I am not sure how to show uniqueness here?

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3) Show the collection of all unit vectors $\displaystyle \in H \cong S^1$ over the Bloch sphere.

My answer:

So we can express the state of a qubit as $\displaystyle |\psi\rangle = e^{i\theta}(cos(\dfrac{\theta}{2})|0\rangle + e^{i(\theta_2-\theta_1)}sin(\dfrac{\theta}{2})|1\rangle)$.

Now to each physical state corresponds to one fiber for $\displaystyle 0 \le \theta \lt 2\pi$, or we can write this as an ordered pair, $\displaystyle {(\beta, \theta)| \beta \: a \: point \: on \: the \: Bloch \: sphere, \: 0 \le \theta \lt 2\pi}$.How do I manipulate this to show the isomorphism?

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4) Show the collection of all state vectors is $\displaystyle \cong \mathbb{C}P^1$.

My answer:

Note that in quantum mechanics$\displaystyle |\psi\rangle$ and $\displaystyle \lambda|\psi\rangle$ represent the same physical state for $\displaystyle \lambda \ne 0$. So here in $\displaystyle H_2, \: |\psi\rangle = \alpha|0\rangle + \beta|1\rangle \equiv |\psi\rangle = \lambda(\alpha|0\rangle + \beta|1\rangle)$. These states in $\displaystyle H$ are parametrized by the pairs of complex numbers $\displaystyle (\alpha,\beta)$. Remember $\displaystyle \mathbb{C}P^1$ is the subset of $\displaystyle \mathbb{C}^2$ consisting of complex pairs $\displaystyle (\alpha,\beta)$ such that $\displaystyle (\alpha,\beta)=\lambda(\alpha,\beta)=(\lambda\alph a,\lambda\beta)$.

What should I be doing from here?