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bures_angle

Bures angle, also known as Bures arc, Bures length or quantum angle is a distance metric.

The Bures angle metric is a measure of the statistical distance between quantum states.

bures_angle 

bures_angle(
    rho_1: ndarray, rho_2: ndarray, decimals: int = 10
) -> float

Compute the Bures angle of two density matrices 1.

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

\[ \arccos{\sqrt{F (\rho_1, \rho_2)}} \]

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

Parameters:

  • rho_1 (ndarray) –

    Density operator.

  • rho_2 (ndarray) –

    Density operator.

  • decimals (int, default: 10 ) –

    Number of decimal places to round to (default 10).

Returns:

  • float –

    The Bures angle between rho_1 and rho_2.

Raises:

  • ValueError –

    If matrices are not of equal dimension.

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

import numpy as np
from toqito.state_metrics import bures_angle

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_angle(rho, sigma))

0.0

References

1 Wikipedia. Bures distance. link.

Source code in toqito/state_metrics/bures_angle.py 
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def bures_angle(rho_1: np.ndarray, rho_2: np.ndarray, decimals: int = 10) -> float:
    r"""Compute the Bures angle of two density matrices [@wikipediabures].

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

    \[
        \arccos{\sqrt{F (\rho_1, \rho_2)}}
    \]

    where \(F(\cdot)\) denotes the fidelity between \(\rho_1\) and \(\rho_2\). The return is a value between
    \(0\) and \(\pi / 2\), with \(0\) corresponding to matrices `rho_1 = rho_2` and \(\pi / 2\)
    corresponding to the case `rho_1` and `rho_2` with orthogonal support.

    Args:
        rho_1: Density operator.
        rho_2: Density operator.
        decimals: Number of decimal places to round to (default 10).

    Returns:
        The Bures angle between `rho_1` and `rho_2`.

    Raises:
        ValueError: If matrices are not of equal dimension.

    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 angle 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" result="text"
        import numpy as np
        from toqito.state_metrics import bures_angle

        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_angle(rho, sigma))
        ```

    """
    # Perform error checking.
    if not np.all(rho_1.shape == rho_2.shape):
        raise ValueError("InvalidDim: `rho_1` and `rho_2` must be matrices of the same size.")
    # Round fidelity to only 10 decimals to avoid error when `rho_1 = rho_2`.
    return np.real(np.arccos(np.sqrt(np.round(fidelity(rho_1, rho_2), decimals))))