state_metrics.hilbert_schmidt_inner_product

Hilbert-Schmidt Inner Product.

Module Contents

Functions

hilbert_schmidt_inner_product(a_mat, b_mat)

Compute the Hilbert-Schmidt inner product between two matrices [2].

state_metrics.hilbert_schmidt_inner_product.hilbert_schmidt_inner_product(a_mat, b_mat)

Compute the Hilbert-Schmidt inner product between two matrices [2].

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 [1].

Examples

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

>>> from toqito.matrices import hadamard
>>> from toqito.matrices import pauli
>>> h = hadamard(1)
>>> hilbert_schmidt_inner_product(h, h)
2

References

[1]

Rigetti. Forest benchmarking. https://github.com/rigetti/forest-benchmarking.

[2] (1,2)

Wikipedia. Hilbert-schmidt operator. https://en.wikipedia.org/wiki/Hilbert.

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