Parametrization of uniformly distributed n dimensional states

[Arubaito]

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

 

[tom.stoer]

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.

 

[Arubaito]

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.

 

[mfb]

The uniform distribution on the spheres gives you this generalization - you just have to add phase factors for all but one base vector afterwards.

 

[andrien]

are you looking for generalization of spherical coordinates then
x₁=rCos(ψ₁)
x₂=rSin(ψ₁)Cos(ψ₂) and so on
with x_n=rSin(ψ₁)...Sin(ψ_n-2)Sin(ψ_n-1)
ψ_n-1 ranges over 0 to 2∏ and other ranges from 0 to ∏.

 

[Arubaito]

I am not very sure that andrien parametrization suffices. Following mfb, dou you think that something of the following form works?

<br /> |\phi\rangle=\cos\frac{\theta_1}{2}|0\rangle <br /> + e^{i\phi_1}\sin\frac{\theta_1}{2}\cos\frac{\theta_2}{2}|1\rangle<br /> + e^{i\phi_2}\sin\frac{\theta_1}{2}\sin\frac{\theta_2}{2}\cos\frac{\theta_3}{2}|2\rangle<br /> + \ldots<br /> + e^{i\phi_{n-2}}\sin\frac{\theta_1}{2}\ldots\cos\frac{\theta_{n-2}}{2}|n-2\rangle<br /> + e^{i\phi_{n-1}}\sin\frac{\theta_1}{2}\ldots\sin\frac{\theta_{n-2}}{2}|n-1\rangle<br />

 

[tom.stoer]

Have a look a the n-dim. spherical coordinates http://en.wikipedia.org/wiki/N-sphere#Spherical_coordinates

 

[Arubaito]

Hi Tom, I took a look, however the parametrization I need should be in the computational basis. What is wrong with:

<br /> |\phi\rangle=\cos\frac{\theta_1}{2}|0\rangle \\<br /> + e^{i\phi_1}\sin\frac{\theta_1}{2}\cos\frac{\theta_2}{2}|1\rangle \\<br /> + e^{i\phi_2}\sin\frac{\theta_1}{2}\sin\frac{\theta_2}{2}\cos\frac{\theta_3}{2}|2\rangle \\<br /> + \ldots \\<br /> + e^{i\phi_{n-2}}\sin\frac{\theta_1}{2}\ldots\cos\frac{\theta_{n-2}}{2}|n-2\rangle \\<br /> + e^{i\phi_{n-1}}\sin\frac{\theta_1}{2}\ldots\sin\frac{\theta_{n-2}}{2}|n-1\rangle<br />

 

[tom.stoer]

Nothing is wrong, but a uniform random distribution of the angles results in a non-uniform distribution on the n-sphere

 

[Arubaito]

I see, you mean that I should consider also the appropriate element of area, which following your pointer should be:

<br /> (\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}<br />

with 0\leq\theta_i\leq\pi and 0\leq\phi_i\leq 2\pi.

 

[tom.stoer]

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

 

[mfb]

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.

 

[Arubaito]

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

 

[tom.stoer]

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

... 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.

 

[Arubaito]

You're right, my question should have been more precise.