toqito.matrix_ops.vectors_to_gram_matrix

Calculates the Gram matrix from a list of vectors.

Module Contents

toqito.matrix_ops.vectors_to_gram_matrix.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)

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.

Parameters:

vectors (list[numpy.ndarray]) – 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.

Return type:

numpy.ndarray