state_metrics.hilbert_schmidt

Hilbert-Schmidt metric.

Module Contents

Functions

hilbert_schmidt(rho, sigma)

Compute the Hilbert-Schmidt distance between two states [2].
state_metrics.hilbert_schmidt.hilbert_schmidt(rho, sigma)

Compute the Hilbert-Schmidt distance between two states [2].

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

\[D_{\text{HS}}(\rho, \sigma) = \text{Tr}((\rho - \sigma)^2) = \left\lVert \rho - \sigma \right\rVert_2^2.\]

Examples

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

>>> 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
>>> '%.2f' % hilbert_schmidt(rho, sigma)
'1.00'

Note

You do not need to use ‘%.2f’ % when you use this function.

We use this to format our output such that doctest compares the calculated output to the expected output upto two decimal points only. The accuracy of the solvers can calculate the float output to a certain amount of precision such that the value deviates after a few digits of accuracy.

References

[1] (1,2)

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

Raises:

ValueError – If matrices are not density operators.

Parameters:

  • rho (numpy.ndarray) – An input matrix.

  • sigma (numpy.ndarray) – An input matrix.

Returns:

The Hilbert-Schmidt distance between rho and sigma.

Return type:

float