Hey ncshields.
Can you please fix up the latex (on my end, all I see is broken links to images)?
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?
The uniqueness means that if you have two points |a> and |b> then basically |a> = |b> if and only if a_lambda = b_lambda and a_mu = b_mu. If this is the case then the points must be unique.
For the last one you must show iso-morphism between the spaces. Have you covered this aspect in class?