toqito.state_metrics.fidelity

toqito.state_metrics.fidelity(rho, sigma)

Compute the fidelity of two density matrices [WikFid].

Calculate the fidelity between the two density matrices rho and sigma, defined by:

\[||\sqrt(\rho) \sqrt(\sigma)||_1,\]

where \(|| \cdot ||_1\) denotes the trace norm. The return is a value between \(0\) and \(1\), with \(0\) corresponding to matrices rho and sigma with orthogonal support, and \(1\) corresponding to the case rho = sigma.

Examples

Consider the following Bell state

\[u = \frac{1}{\sqrt{2}} \left( |00 \rangle + |11 \rangle \right) \in \mathcal{X}.\]

The corresponding density matrix of \(u\) may be calculated by:

\[\begin{split}\rho = u u^* = \frac{1}{2} \begin{pmatrix} 1 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 1 \end{pmatrix} \in \text{D}(\mathcal{X}).\end{split}\]

In the event where we calculate the fidelity between states that are identical, we should obtain the value of \(1\). This can be observed in toqito as follows.

>>> from toqito.state_metrics import fidelity
>>> import numpy as np
>>> rho = 1 / 2 * np.array(
>>>     [[1, 0, 0, 1],
>>>      [0, 0, 0, 0],
>>>      [0, 0, 0, 0],
>>>      [1, 0, 0, 1]]
>>> )
>>> sigma = rho
>>> fidelity(rho, sigma)
1.0000000000000002

References

[WikFid]

Wikipedia: Fidelity of quantum states https://en.wikipedia.org/wiki/Fidelity_of_quantum_states

Raises:

ValueError – If matrices are not density operators.

Parameters:

• rho – Density operator.

• sigma – Density operator.

Returns:

The fidelity between rho and sigma.