toqito.state_metrics.bures_distance

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

Compute the Bures distance of two density matrices [WikBures].

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:

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

In the event where we calculate the Bures distance between states that are identical, we should obtain the value of \(0\). This can be observed in toqito as follows.

>>> from toqito.state_metrics import bures_distance
>>> 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
>>> bures_distance(rho, sigma)
0

References

Raises:

ValueError – If matrices are not of equal dimension.

Parameters:

  • rho_1 – Density operator.

  • rho_2 – Density operator.

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

Returns:

The Bures distance between rho_1 and rho_2.