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

1) Let be a unit vector in a hilbert space, . So and . Show that we can rotate so that the coefficient of is real and nonnegative.

My answer:

Now take . Then . Now rotate it through by .

.

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

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2) And let . Show that each such vector in 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 over the Bloch sphere.

My answer:

So we can express the state of a qubit as .

Now to each physical state corresponds to one fiber for , or we can write this as an ordered pair, .How do I manipulate this to show the isomorphism?

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4) Show the collection of all state vectors is .

My answer:

Note that in quantum mechanics and represent the same physical state for . So here in . These states in are parametrized by the pairs of complex numbers . Remember is the subset of consisting of complex pairs such that .

What should I be doing from here?