toqito.state_metrics.fidelity

Fidelity is a metric that qualifies how close two quantum states are.

Module Contents

toqito.state_metrics.fidelity.fidelity(rho, sigma)

Compute the fidelity of two density matrices [@WikiFidQuant].

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:

[

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}).

]

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.

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

rho = 1 / 2 * np.array(

[[1, 0, 0, 1],

[0, 0, 0, 0], [0, 0, 0, 0], [1, 0, 0, 1]]

) sigma = rho

print(fidelity(rho, sigma)) ```

Raises:

ValueError – If matrices are not density operators.

Parameters:

• rho (numpy.ndarray) – Density operator.

• sigma (numpy.ndarray) – Density operator.

Returns:

The fidelity between rho and sigma.

Return type:

float