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sub_fidelity

Sub-fidelity metric is a lower bound for the fidelity.

The sub-fidelity metric is a concave function and sub-multiplicative.

sub_fidelity 

sub_fidelity(rho: ndarray, sigma: ndarray) -> float

Compute the sub fidelity of two density matrices 1.

The sub-fidelity is a measure of similarity between density operators. It is defined as

\[ E(\rho, \sigma) = \text{Tr}(\rho \sigma) + \sqrt{2 \left[ \text{Tr}(\rho \sigma)^2 - \text{Tr}(\rho \sigma \rho \sigma) \right]}, \]

where \(\sigma\) and \(\rho\) are density matrices. The sub-fidelity serves as a lower bound for the fidelity.

Parameters:

  • rho (ndarray) –

    Density operator.

  • sigma (ndarray) –

    Density operator.

Returns:

  • float –

    The sub-fidelity between rho and sigma.

Raises:

  • ValueError –

    If matrices are not of equal dimension.

Examples:

Consider the following pair of states:

\[ \rho = \frac{3}{4}|0\rangle \langle 0| + \frac{1}{4}|1 \rangle \langle 1| \quad \text{and} \quad \sigma = \frac{1}{8}|0 \rangle \langle 0| + \frac{7}{8}|1 \rangle \langle 1|. \]

Calculating the fidelity between the states \(\rho\) and \(\sigma\) as \(F(\rho, \sigma) \approx 0.774\). This can be observed in |toqito⟩ as

from toqito.states import basis
from toqito.state_metrics import fidelity

e_0, e_1 = basis(2, 0), basis(2, 1)
rho = 3 / 4 * e_0 @ e_0.conj().T + 1 / 4 * e_1 @ e_1.conj().T
sigma = 1/8 * e_0 @ e_0.conj().T + 7/8 * e_1 @ e_1.conj().T

print(fidelity(rho, sigma))

0.77389339119464

As the sub-fidelity is a lower bound on the fidelity, that is \(E(\rho, \sigma) \leq F(\rho, \sigma)\), we can use |toqito⟩ to observe that \(E(\rho, \sigma) \approx 0.599 \leq F(\rho, \sigma) \approx 0.774\).

from toqito.states import basis
from toqito.state_metrics import sub_fidelity

e_0, e_1 = basis(2, 0), basis(2, 1)
rho = 3 / 4 * e_0 @ e_0.conj().T + 1 / 4 * e_1 @ e_1.conj().T
sigma = 1/8 * e_0 @ e_0.conj().T + 7/8 * e_1 @ e_1.conj().T

print(sub_fidelity(rho, sigma))

0.5989109809347399

References

1 Miszczak, J. and Puchała, Z. and Horodecki, P. and Uhlmann, A. and Życzkowski, K.. Sub-- and super--fidelity as bounds for quantum fidelity. (2008).

Source code in toqito/state_metrics/sub_fidelity.py 
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def sub_fidelity(rho: np.ndarray, sigma: np.ndarray) -> float:
    r"""Compute the sub fidelity of two density matrices [@miszczak2008sub].

    The sub-fidelity is a measure of similarity between density operators. It is defined as

    \[
        E(\rho, \sigma) = \text{Tr}(\rho \sigma) +
        \sqrt{2 \left[ \text{Tr}(\rho \sigma)^2 - \text{Tr}(\rho \sigma \rho \sigma) \right]},
    \]

    where \(\sigma\) and \(\rho\) are density matrices. The sub-fidelity serves as a lower bound for the
    fidelity.

    Args:
        rho: Density operator.
        sigma: Density operator.

    Returns:
        The sub-fidelity between `rho` and `sigma`.

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

    Examples:
        Consider the following pair of states:

        \[
            \rho = \frac{3}{4}|0\rangle \langle 0| +
                   \frac{1}{4}|1 \rangle \langle 1|
                    \quad \text{and} \quad
            \sigma = \frac{1}{8}|0 \rangle \langle 0| +
                     \frac{7}{8}|1 \rangle \langle 1|.
        \]

        Calculating the fidelity between the states \(\rho\) and \(\sigma\) as \(F(\rho, \sigma) \approx
        0.774\). This can be observed in `|toqito⟩` as

        ```python exec="1" source="above" result="text"
        from toqito.states import basis
        from toqito.state_metrics import fidelity

        e_0, e_1 = basis(2, 0), basis(2, 1)
        rho = 3 / 4 * e_0 @ e_0.conj().T + 1 / 4 * e_1 @ e_1.conj().T
        sigma = 1/8 * e_0 @ e_0.conj().T + 7/8 * e_1 @ e_1.conj().T

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

        As the sub-fidelity is a lower bound on the fidelity, that is \(E(\rho, \sigma) \leq F(\rho, \sigma)\), we can
        use `|toqito⟩` to observe that \(E(\rho, \sigma) \approx 0.599 \leq F(\rho, \sigma) \approx 0.774\).

        ```python exec="1" source="above" result="text"
        from toqito.states import basis
        from toqito.state_metrics import sub_fidelity

        e_0, e_1 = basis(2, 0), basis(2, 1)
        rho = 3 / 4 * e_0 @ e_0.conj().T + 1 / 4 * e_1 @ e_1.conj().T
        sigma = 1/8 * e_0 @ e_0.conj().T + 7/8 * e_1 @ e_1.conj().T

        print(sub_fidelity(rho, sigma))
        ```

    """
    # Perform some error checking.
    if not np.all(rho.shape == sigma.shape):
        raise ValueError("InvalidDim: `rho` and `sigma` must be matrices of the same size.")
    if not is_density(rho) or not is_density(sigma):
        raise ValueError("Sub-fidelity is only defined for density operators.")

    inner = np.trace(rho @ sigma)
    # The radicand is non-negative in exact arithmetic, but floating-point error can push it
    # slightly below zero for pure/near-equal states; clamp it before the square root.
    radicand = 2 * (inner**2 - np.trace(rho @ sigma @ rho @ sigma))
    return np.real(inner + np.sqrt(np.maximum(np.real(radicand), 0.0)))