toqito.matrix_ops.vectors_to_gram_matrix
========================================

.. py:module:: toqito.matrix_ops.vectors_to_gram_matrix

.. autoapi-nested-parse::

   Calculates the Gram matrix from a list of vectors.





Module Contents
---------------

.. py:function:: vectors_to_gram_matrix(vectors)

   Construct the Gram matrix from a list of vectors or density matrices [@WikiGram].

   The Gram matrix is a matrix of inner products. This function automatically detects whether the inputs
   are vectors (pure states) or density matrices (mixed states) and computes the appropriate Gram matrix.

   For vectors |ψᵢ⟩: G[i, j] = ⟨ψᵢ|ψⱼ⟩
   For density matrices ρᵢ: G[i, j] = Tr(ρᵢ ρⱼ)

   .. rubric:: Examples

   Example with real vectors:

   ```python exec="1" source="above"
   import numpy as np
   from toqito.matrix_ops import vectors_to_gram_matrix

   vectors = [np.array([1, 2]), np.array([3, 4])]
   gram_matrix = vectors_to_gram_matrix(vectors)

   print(gram_matrix)
   ```

   Example with complex vectors:

   ```python exec="1" source="above"
   import numpy as np
   from toqito.matrix_ops import vectors_to_gram_matrix

   vectors = [np.array([1+1j, 2+2j]), np.array([3+3j, 4+4j])]
   gram_matrix = vectors_to_gram_matrix(vectors)

   print(gram_matrix)
   ```

   Example with density matrices (mixed states):

   ```python exec="1" source="above"
   import numpy as np
   from toqito.matrix_ops import vectors_to_gram_matrix

   # Two mixed states
   rho1 = 0.7 * np.array([[1., 0.], [0., 0.]]) + 0.3 * np.eye(2) / 2
   rho2 = 0.7 * np.array([[0., 0.], [0., 1.]]) + 0.3 * np.eye(2) / 2
   states = [rho1, rho2]

   gram_matrix = vectors_to_gram_matrix(states)
   print(gram_matrix)
   ```

   :raises ValueError: If the vectors are not all of the same shape.

   :param vectors: A list of vectors (1D/column arrays for pure states) or density matrices (2D arrays for mixed states).

   :returns: The Gram matrix with entries G[i,j] = ⟨vᵢ|vⱼ⟩ for vectors or Tr(ρᵢρⱼ) for density matrices.