Random orthonormal basis in matrix_props

docs/refs.bib

@@ -806,6 +806,15 @@ @misc{Russo_2017_Extended
 }
 
 #Last name begins with S
+
+@misc{SE_1688950,
+  author  = "Stack Exchange Mathematics",
+  title   = "Why do the columns of a unitary matrix form an orthonormal basis?",
+  howpublished = {https://math.stackexchange.com/q/1688950}
+
+}
+
+
 @misc{Seshadri_2021_Git,
   author  = "Seshadri, Akshay",
   title   = "Minimax Fidelity Estimation",

toqito/matrix_props/is_orthonormal.py

@@ -39,4 +39,5 @@ def is_orthonormal(vectors: list[np.ndarray]) -> bool:
     :return: True if vectors are orthonormal; False otherwise.
 
     """
-    return is_mutually_orthogonal(vectors) and np.allclose(np.dot(vectors, vectors.T), np.eye(vectors.shape[0]))
+    return is_mutually_orthogonal(vectors) and np.allclose(np.dot(vectors, np.conjugate(vectors).T), np.eye(
+        vectors.shape[0]))

toqito/rand/__init__.py

@@ -7,3 +7,4 @@
 from toqito.rand.random_state_vector import random_state_vector
 from toqito.rand.random_states import random_states
 from toqito.rand.random_circulant_gram_matrix import random_circulant_gram_matrix
+from toqito.rand.random_orthonormal_basis import random_orthonormal_basis

toqito/rand/random_orthonormal_basis.py

@@ -0,0 +1,40 @@
+"""Generate random orthonormal basis."""
+import numpy as np
+
+from toqito.matrix_props import is_orthonormal
+from toqito.rand import random_unitary
+
+
+def random_orthonormal_basis(dim: int, is_real: bool = False) -> list[np.ndarray]:
+    r"""Generate a real random orthonormal basis of given dimension :math:`d`.
+
+    The basis is generated from the columns of a random unitary matrix of the same dimension
+    as the columns of a unitary matrix typically form an orthonormal basis :cite:`SE_1688950`.
+
+    Examples
+    ==========
+    To generate a random orthonormal basis of dimension :math:`4`,
+
+    >>> from toqito.rand import random_orthonormal_basis
+    >>> random_orthonormal_basis(4, is_real = True) # doctest: +SKIP
+    [array([0.52188745, 0.4983613 , 0.69049811, 0.04981832]),
+    array([-0.48670459,  0.58756912, -0.10226756,  0.63829658]),
+    array([ 0.23965404, -0.58538248,  0.187136  ,  0.75158061]),
+    array([ 0.658269  ,  0.25243989, -0.69118291,  0.158815  ])]
+
+    References
+    ==========
+    .. bibliography::
+        :filter: docname in docnames
+
+    dim: int
+        Number of elements in the random orthonormal basis.
+
+    """
+    random_mat = random_unitary(dim, is_real)
+
+    rand_orth_basis = [random_mat[:, i] for i in range(dim)]
+
+    return rand_orth_basis
+
+

toqito/rand/tests/test_random_orthonormal_basis.py

@@ -0,0 +1,17 @@
+"""Tests for random orthonormal basis."""
+
+import numpy as np
+import pytest
+
+from toqito.matrix_props import is_orthonormal
+from toqito.rand import random_orthonormal_basis
+
+
+@pytest.mark.parametrize("input_dim", range(2, 5))
+@pytest.mark.parametrize("bool", [False, True])
+def test_random_orth_basis_int_dim(input_dim, bool):
+    """Test function works as expected for a valid int input."""
+    gen_basis = random_orthonormal_basis(dim=input_dim, is_real = bool)
+    assert len(gen_basis) == input_dim
+    assert is_orthonormal(np.array(gen_basis))
+