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Math Help - Operating in Hilbert Spaces - Isomorphisms

  1. #1
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    Operating in Hilbert Spaces - Isomorphisms

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


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


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

    |\phi\rangle = r_1e^{i\theta_1}e^{-i\theta_1}|0\rangle + r_2e^{i\theta_2}e^{-i\theta_1}|1\rangle
    = r_1e^0|0\rangle + r_2e^{i\theta_2-i\theta_1}|1\rangle
    = 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 |\omega\rangle = \lambda'|0\rangle + \mu'|1\rangle; \: \lambda',\mu' \in \mathbb{R}_{\ge 0}. Show that each such vector in 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 \in H \cong S^1 over the Bloch sphere.


    My answer:
    So we can express the state of a qubit as |\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 0 \le \theta \lt 2\pi, or we can write this as an ordered pair, {(\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 \cong \mathbb{C}P^1.


    My answer:
    Note that in quantum mechanics  |\psi\rangle and \lambda|\psi\rangle represent the same physical state for \lambda \ne 0. So here in H_2, \: |\psi\rangle = \alpha|0\rangle + \beta|1\rangle \equiv |\psi\rangle = \lambda(\alpha|0\rangle + \beta|1\rangle). These states in H are parametrized by the pairs of complex numbers (\alpha,\beta). Remember \mathbb{C}P^1 is the subset of \mathbb{C}^2 consisting of complex pairs (\alpha,\beta) such that (\alpha,\beta)=\lambda(\alpha,\beta)=(\lambda\alph  a,\lambda\beta).
    What should I be doing from here?
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  2. #2
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    Re: Operating in Hilbert Spaces - Isomorphisms

    Hey ncshields.

    Can you please fix up the latex (on my end, all I see is broken links to images)?
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  3. #3
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    Re: Operating in Hilbert Spaces - Isomorphisms

    Quote Originally Posted by chiro View Post
    Hey ncshields.

    Can you please fix up the latex (on my end, all I see is broken links to images)?
    It was not working on my end either, but I just shut the window and reopened mathhelpforum and the latex was fine. Is it working for you now?

    You might also try a different browser.
    Last edited by ncshields; February 16th 2014 at 07:24 PM.
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  4. #4
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    Re: Operating in Hilbert Spaces - Isomorphisms

    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?
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  5. #5
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    Re: Operating in Hilbert Spaces - Isomorphisms

    Quote Originally Posted by chiro View Post
    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?
    Ok that definitely helps. Well part 2 is a large part of showing injectivity in part 3, but I am not sure how to do the surjective part?
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  6. #6
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    Re: Operating in Hilbert Spaces - Isomorphisms

    The first thing in any group theoretic exercise is to write down the sets involved and the binary operation. Given your question, can you identify these from the last question statement?
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