state_metrics.hilbert_schmidt_inner_product

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

Functions

hilbert_schmidt_inner_product(a_mat, b_mat)

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

Module Contents

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.

>>> import numpy as np
>>> from toqito.matrices import hadamard
>>> from toqito.state_metrics import hilbert_schmidt_inner_product
>>> h = hadamard(1)
>>> np.around(hilbert_schmidt_inner_product(h, h), decimals=2)
np.float64(2.0)

References

[1]

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

[2] (1,2)

Wikipedia. Hilbert-schmidt operator. URL: https://en.wikipedia.org/wiki/Hilbert%E2%80%93Schmidt_operator.

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