Parametrization of uniformly distributed n dimensional states

In summary: Thank you for your help!In summary, the uniform distribution on the spheres gives you a generalization of spherical coordinates which can be used to integrate over the space of states.
  • #1
Arubaito
8
0
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?
 
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  • #2
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:
  • #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.
 
  • #4
The uniform distribution on the spheres gives you this generalization - you just have to add phase factors for all but one base vector afterwards.
 
  • #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 ∏.
 
  • #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
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]
 
  • #9
Nothing is wrong, but a uniform random distribution of the angles results in a non-uniform distribution on the n-sphere
 
  • #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].
 
  • #11
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
 
  • #12
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.
 
  • #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
 
  • #14
Arubaito said:
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
Right.

But in the 1st post you wrote

Arubaito said:
... To pick one such tate uniformly at random it suffices to draw ... at random from its domain and ... uniformly in the range ... How would you do the equivalent parametrization for an n-dimensional state?

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

1. What is parametrization of uniformly distributed n dimensional states?

Parametrization of uniformly distributed n dimensional states is a method used to describe a set of objects or states in n-dimensional space in a systematic way. It involves assigning a set of parameters to each state, which allows for easy identification and comparison of the states.

2. Why is parametrization of uniformly distributed n dimensional states important?

Parametrization of uniformly distributed n dimensional states is important because it allows for a more efficient and organized way of describing a large set of objects or states. It also enables easier analysis and comparison of the states, making it a valuable tool in many scientific fields.

3. How is parametrization of uniformly distributed n dimensional states used in scientific research?

Parametrization of uniformly distributed n dimensional states is used in scientific research in various fields such as physics, chemistry, and statistics. It is commonly used to describe the states of particles, molecules, or data points in a dataset. It is also used in statistical modeling and simulation studies.

4. What are the advantages of using parametrization of uniformly distributed n dimensional states?

There are several advantages to using parametrization of uniformly distributed n dimensional states. It allows for a compact and organized representation of a large set of states. It also makes it easier to perform statistical analysis and comparison of the states. Additionally, it can be used to simplify complex systems and make them more manageable for further study.

5. Can parametrization of uniformly distributed n dimensional states be applied to non-uniformly distributed states?

Yes, parametrization of uniformly distributed n dimensional states can be applied to non-uniformly distributed states. In this case, the parameters may not be equally spaced, but they can still be used to describe and compare the states in a systematic way. However, this approach may not be as effective as in the case of uniformly distributed states.

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