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Parametrization of uniformly distributed n dimensional states

  1. May 1, 2013 #1
    Any two dimensional state can be written as:
    [itex]
    |\phi\rangle=\cos\frac{\theta}{2}|0\rangle+e^{i\phi}\sin\frac{\theta}{2}|1\rangle
    [/itex]
    where [itex]0\leq\theta\leq\pi[/itex] and [itex]0\leq\phi\leq 2\pi[/itex], and [itex]0\leq\theta\leq\pi[/itex]. To pick one such state uniformly at random it suffices to draw [itex]\phi[/itex] at random from its domain and [itex]\cos\theta[/itex] uniformly in the range [itex][-1,1] [/itex]. How would you do the equivalent parametrization for an n-dimensional state?
     
  2. jcsd
  3. May 1, 2013 #2

    tom.stoer

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    I think you should google for "random distribution on n-spheres"

    It is not possible to use n-dim. spherical coordinates; the sample will not be distributed uniformly. However it is possible to generate a random point X uniformly distributed in the n-cube [−1,1]n, to reject it if it is outside the unit ball |X| ≤ 1, and to project the remaining points to the n-sphere. This works nice for small n, but fails for large n b/c the ratio Vol(n-sphere) / Vol(n-cube) tends to zero for large n.
     
    Last edited: May 1, 2013
  4. May 1, 2013 #3
    Thanks for your pointer. Probably my question was a little ambiguous. The thing is that I want to use the parametrization to integrate over the space of states, that's why I was looking for a generalization of spherical coordinates.
     
  5. May 1, 2013 #4

    mfb

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    The uniform distribution on the spheres gives you this generalization - you just have to add phase factors for all but one base vector afterwards.
     
  6. May 1, 2013 #5
    are you looking for generalization of spherical coordinates then
    x1=rCos(ψ1)
    x2=rSin(ψ1)Cos(ψ2) and so on
    with xn=rSin(ψ1)....Sin(ψn-2)Sin(ψn-1)
    ψn-1 ranges over 0 to 2∏ and other ranges from 0 to ∏.
     
  7. May 1, 2013 #6
    I am not very sure that andrien parametrization suffices. Following mfb, dou you think that something of the following form works?

    [itex]
    |\phi\rangle=\cos\frac{\theta_1}{2}|0\rangle
    + e^{i\phi_1}\sin\frac{\theta_1}{2}\cos\frac{\theta_2}{2}|1\rangle
    + e^{i\phi_2}\sin\frac{\theta_1}{2}\sin\frac{\theta_2}{2}\cos\frac{\theta_3}{2}|2\rangle
    + \ldots
    + e^{i\phi_{n-2}}\sin\frac{\theta_1}{2}\ldots\cos\frac{\theta_{n-2}}{2}|n-2\rangle
    + e^{i\phi_{n-1}}\sin\frac{\theta_1}{2}\ldots\sin\frac{\theta_{n-2}}{2}|n-1\rangle
    [/itex]
     
  8. May 1, 2013 #7

    tom.stoer

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  9. May 13, 2013 #8
    Hi Tom, I took a look, however the parametrization I need should be in the computational basis. What is wrong with:

    [itex]
    |\phi\rangle=\cos\frac{\theta_1}{2}|0\rangle \\
    + e^{i\phi_1}\sin\frac{\theta_1}{2}\cos\frac{\theta_2}{2}|1\rangle \\
    + e^{i\phi_2}\sin\frac{\theta_1}{2}\sin\frac{\theta_2}{2}\cos\frac{\theta_3}{2}|2\rangle \\
    + \ldots \\
    + e^{i\phi_{n-2}}\sin\frac{\theta_1}{2}\ldots\cos\frac{\theta_{n-2}}{2}|n-2\rangle \\
    + e^{i\phi_{n-1}}\sin\frac{\theta_1}{2}\ldots\sin\frac{\theta_{n-2}}{2}|n-1\rangle
    [/itex]
     
  10. May 13, 2013 #9

    tom.stoer

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    Nothing is wrong, but a uniform random distribution of the angles results in a non-uniform distribution on the n-sphere
     
  11. May 16, 2013 #10
    I see, you mean that I should consider also the appropriate element of area, which following your pointer should be:

    [itex]
    (\sin{\theta_1})^{n-2}d\theta_1(\sin{\theta_2})^{n-3}d\theta_2 \ldots d\theta_{n-2} d\phi_1 d\phi_2\ldots d\phi_{n-1}
    [/itex]

    with [itex]0\leq\theta_i\leq\pi[/itex] and [itex]0\leq\phi_i\leq 2\pi[/itex].
     
  12. May 16, 2013 #11

    tom.stoer

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    yes

    but honerstly, I do not know how to generate random angles with a specific non-uniform distribution such that the resulting points on the sphere are uniformely distributed. Better try a google search
     
  13. May 16, 2013 #12

    mfb

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    Staff: Mentor

    Generate n random numbers from -1 to 1, interpret them as vector, normalize? Calculate the vectors afterwards if you like.

    To avoid numerical issues, discard vectors with a very small magnitude due to rounding errors.
     
  14. May 16, 2013 #13
    What you're saying helps if I wanted to sample vectors, but I think it doesn't help me much if I want to manipulate analitically the state
     
  15. May 16, 2013 #14

    tom.stoer

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    Right.

    But in the 1st post you wrote

    So my conclusion was that you are looking for random samples on th n-dim. unit sphere.
     
  16. May 16, 2013 #15
    You're right, my question should have been more precise.
     
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