Random orthonormal basis in matrix_props

[purva-thakre]

Description

Fixes #435

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@@          Coverage Diff           @@
##           master    #821   +/-   ##
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  Coverage    97.8%   97.8%           
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  Files         166     168    +2     
  Lines        3242    3261   +19     
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[purva-thakre]

when is_real = False, perhaps is_orthonormal requires a tolerance?

toqito/toqito/matrix_props/is_orthonormal.py

Line 42 in ac02365

return is_mutually_orthogonal(vectors) and np.allclose(np.dot(vectors, vectors.T), np.eye(vectors.shape[0]))

[purva-thakre]

@vprusso As discussed, here's the minimum working example when is_real=False in random_unitary. The created basis elements are mutually orthogonal but they fail at the second check.

np.dot on L42 of is_orthonormal is not creating a matrix close to np.eye. I don't think .T works that well when the matrix elements are complex.

import numpy as np
from toqito.rand import random_unitary
from toqito.matrix_props import is_orthonormal, is_unitary
from toqito.state_props import is_mutually_orthogonal

dim = 4
random_mat = random_unitary(dim, is_real=False)

if is_unitary(random_mat):
    print("Random matrix is unitary.")

rand_orth_basis = [random_mat[:, i] for i in range(dim)]

if is_mutually_orthogonal(rand_orth_basis):
    print("The generated basis elements are mutually orthogonal.")

if not is_orthonormal(np.array(rand_orth_basis)):
    raise ValueError("Random orthonormal basis could not constructed.")

Does not work

np.dot(np.array(rand_orth_basis), np.array(rand_orth_basis).T)

Works

np.dot(np.array(rand_orth_basis), np.conjugate(np.array(rand_orth_basis)).T)

[vprusso]

Minor comment, but aside from that approved! LGTM!