toqito.state_metrics.bures_distance

Bures distance metric is a commonly used distance metric.

It serves as an actual measure of distinguishability between two quantum states.

Module Contents

toqito.state_metrics.bures_distance.bures_distance(rho_1, rho_2, decimals=10)

Compute the Bures distance of two density matrices [@WikiBures].

Calculate the Bures distance between two density matrices rho_1 and rho_2 defined by:

[

sqrt{2 (1 - F(rho_1, rho_2))},

]

where (F(cdot)) denotes the fidelity between (rho_1) and (rho_2). The return is a value between (0) and (sqrt{2}),with (0) corresponding to matrices: rho_1 = rho_2 and (sqrt{2}) corresponding to the case: rho_1 and rho_2 with orthogonal support.

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 Bures distance between states that are identical, we should obtain the value of (0). This can be observed in |toqito⟩ as follows.

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

rho = 1 / 2 * np.array(

[[1, 0, 0, 1],

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

) sigma = rho

print(bures_distance(rho, sigma)) ```

Raises:

ValueError – If matrices are not of equal dimension.

Parameters:

• rho_1 (numpy.ndarray) – Density operator.

• rho_2 (numpy.ndarray) – Density operator.

• decimals (int) – Number of decimal places to round to (default 10).

Returns:

The Bures distance between rho_1 and rho_2.

Return type:

float