toqito.state_metrics.hilbert_schmidt¶

Hilbert-Schmidt metric is a distance metric used to generate an entanglement measure.

Module Contents¶

toqito.state_metrics.hilbert_schmidt.hilbert_schmidt(rho, sigma)[source]¶

Compute the Hilbert-Schmidt distance between two states [@WikiHilbSchOp].

The Hilbert-Schmidt distance between density operators (rho) and (sigma) is defined as

[: D_{text{HS}}(rho, sigma) = text{Tr}((rho - sigma)^2) = leftlVert rho - sigma rightrVert_2^2.

]

Examples

One may consider taking the Hilbert-Schmidt distance between two Bell states. In |toqito⟩, one may accomplish this as

```python exec=”1” source=”above” import numpy as np from toqito.states import bell from toqito.state_metrics import hilbert_schmidt

rho = bell(0) @ bell(0).conj().T sigma = bell(3) @ bell(3).conj().T

print(np.around(hilbert_schmidt(rho, sigma), decimals=2)) ```

Raises:

ValueError – If matrices are not density operators.

Parameters:

Returns:

The Hilbert-Schmidt distance between rho and sigma.

Return type:

float | numpy.floating