toqito.state_metrics.hilbert_schmidt_inner_product¶

Hilbert-Schmidt Inner Product refers to the inner product between two Hilbert-Schmidt operators.

Module Contents¶

toqito.state_metrics.hilbert_schmidt_inner_product.hilbert_schmidt_inner_product(a_mat, b_mat)[source]¶

Compute the Hilbert-Schmidt inner product between two matrices [@WikiHilbSchOp].

The Hilbert-Schmidt inner product between a_mat and b_mat is defined as

[: HS = (A|B) = Tr[A^dagger B]

]

where (|B\rangle = \text{vec}(B)\) and \(\langle A|) is the dual vector to (|A rangle).

Note: This function has been adapted from [@Rigetti_2022_Forest].

Examples

One may consider taking the Hilbert-Schmidt distance between two Hadamard matrices.

```python exec=”1” source=”above” import numpy as np from toqito.matrices import hadamard from toqito.state_metrics import hilbert_schmidt_inner_product

h = hadamard(1)

print(np.around(hilbert_schmidt_inner_product(h, h), decimals=2)) ```

Parameters:

a_mat (numpy.ndarray) – An input matrix provided as a numpy array.
b_mat (numpy.ndarray) – An input matrix provided as a numpy array.

Returns:

The Hilbert-Schmidt inner product between a_mat and b_mat.

Return type:

complex