Add Schmidt Decomposition to `quantum_info`

[diemilio]

What should we add?

Being able to get the Schmidt decomposition of a bipartate state can be useful to determine its degree of entanglement, find the shared eigenvalues of its reduced subsystems, and to compare if two states are equivalent to each other (i.e., they're equiv if they share the same decomposition). It would be great if there was a function under utils, or if the Statevector class had a method to do this for a given state.

The steps to implement should be fairly straight-forward:

1. Reshape the statevector column-vector into a matrix C of dims of (first subsystem) x (second subsystem). (need to consider the case when indices of subsystem are not sequential).
2. Calculate the SVD of that matrix C = UΣVᴴ.
3. The columns of U are the Schmidt eigenvectors |uᵢ⟩ of the first subsystem, the columns of V the eigenvectors |vᵢ⟩ of the second subsystem, the diagonal elems of Σ are the Schmidt coeffs λᵢ.

The function could return a list/array with each (λᵢ, |uᵢ⟩, |vᵢ⟩). Since |uᵢ⟩ and |vᵢ⟩ are normalized statevectors, they could be Statevector objects.

[diemilio]

I'm willing to work on it if there is interest.

[ikkoham]

Thank you for your nice proposal. I agree with this direction. I think that utils is fine. If you open the PR, I'll review it and if you have questions please contact me.

[diemilio]

Thank you so much @ikkoham. I have had a working function for a little while now, so I thought integrating it into qiskit wouldn't be too difficult. However, I started looking into this yesterday and realized I still need a couple of things:

• [x] Make sure it works for the case where the indices of subsystem are not sequential. It seems like the index order of state tensors in qiskit is reversed from what I thought, so I am having a bit of trouble getting the right permutations.
• [x] Make sure the right dimensions are assigned to each of the singular vectors after decomposition. I had only accounted for bipartite systems where all subsystems are of the same dimension (qudits), but the general case is slightly more complicated.

I am currently working on getting these fixed and will submit a PR right after. I will also let you know if I have any questions.

Thanks!

[ikkoham]

Thanks. Yes Qiskit is a little endian, that is the reverse order of usual textbook.

[diemilio]

Completed in #10104