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channel_fidelity

Computes the channel fidelity between two quantum channels.

channel_fidelity 

channel_fidelity(
    choi_1: ndarray, choi_2: ndarray, eps: float = 1e-07
) -> float

Compute the channel fidelity between two quantum channels 1.

Let \(\Phi : \text{L}(\mathcal{Y}) \rightarrow \text{L}(\mathcal{X})\) and \(\Psi: \text{L}(\mathcal{Y}) \rightarrow \text{L}(\mathcal{X})\) be quantum channels. Then the root channel fidelity defined as

\[ \sqrt{F}(\Phi, \Psi) := \text{inf}_{\rho} \sqrt{F}(\Phi(\rho), \Psi(\rho)) \]

where \(\rho \in \text{D}(\mathcal{Z} \otimes \mathcal{X})\) can be calculated by means of the following semidefinite program (Proposition 50) in 1,

\[ \begin{align*} \text{maximize:} \quad & \lambda \\ \text{subject to:} \quad & \lambda \mathbb{I}_{\mathcal{Z}} \leq \text{Re}\left( \text{Tr}_{\mathcal{Y}} \left( Q \right) \right),\\ & \begin{pmatrix} J(\Phi) & Q^* \\ Q & J(\Psi) \end{pmatrix} \geq 0 \end{align*} \]

where \(Q \in \text{L}(\mathcal{Z} \otimes \mathcal{X})\).

Parameters:

  • choi_1 (ndarray) –

    The Choi matrix of the first quantum channel.

  • choi_2 (ndarray) –

    The Choi matrix of the second quantum channel.

  • eps (float, default: 1e-07 ) –

    The solver tolerance for convergence to feasability.

Returns:

  • float –

    The channel fidelity between the channels specified by the quantum channels corresponding to the Choi matrices

  • float –

    choi_1 and choi_2.

Raises:

  • ValueError –

    If matrices are not of equal dimension.

  • ValueError –

    If matrices are not square.

Examples:

For two identical channels, we should expect that the channel fidelity should yield a value of \(1\).

import numpy as np
from toqito.channels import dephasing
from toqito.channel_metrics import channel_fidelity
# The Choi matrices of dimension-4 for the dephasing channel
choi_1 = dephasing(4)
choi_2 = dephasing(4)
print(channel_fidelity(choi_1, choi_2))

0.9999999999309462

We can also compute the channel fidelity between two different channels. For example, we can compute the channel fidelity between the dephasing and depolarizing channels.

import numpy as np
from toqito.channels import dephasing, depolarizing
from toqito.channel_metrics import channel_fidelity
# The Choi matrices of dimension-4 for the dephasing and depolarizing channels
choi_1 = dephasing(4)
choi_2 = depolarizing(4)
print(channel_fidelity(choi_1, choi_2))

0.5003013053611017

References

1 Katariya, Vishal and Wilde, Mark. Geometric distinguishability measures limit quantum channel estimation and discrimination. Quantum Information Processing. vol. 20(2). (2021). doi:10.1007/s11128-021-02992-7.

Source code in toqito/channel_metrics/channel_fidelity.py 
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def channel_fidelity(choi_1: np.ndarray, choi_2: np.ndarray, eps: float = 1e-7) -> float:
    r"""Compute the channel fidelity between two quantum channels [@katariya2021geometric].

    Let \(\Phi : \text{L}(\mathcal{Y}) \rightarrow \text{L}(\mathcal{X})\) and
    \(\Psi: \text{L}(\mathcal{Y}) \rightarrow \text{L}(\mathcal{X})\) be quantum channels. Then
    the root channel fidelity defined as

    \[
        \sqrt{F}(\Phi, \Psi) := \text{inf}_{\rho} \sqrt{F}(\Phi(\rho), \Psi(\rho))
    \]

    where \(\rho \in \text{D}(\mathcal{Z} \otimes \mathcal{X})\) can be calculated by means of
    the following semidefinite program (Proposition 50) in [@katariya2021geometric],

    \[
        \begin{align*}
            \text{maximize:} \quad & \lambda \\
            \text{subject to:} \quad & \lambda \mathbb{I}_{\mathcal{Z}} \leq
                \text{Re}\left( \text{Tr}_{\mathcal{Y}} \left( Q \right) \right),\\
                & \begin{pmatrix}
                    J(\Phi) & Q^* \\
                    Q & J(\Psi)
                \end{pmatrix} \geq 0
        \end{align*}
    \]

    where \(Q \in \text{L}(\mathcal{Z} \otimes \mathcal{X})\).

    Args:
        choi_1: The Choi matrix of the first quantum channel.
        choi_2: The Choi matrix of the second quantum channel.
        eps: The solver tolerance for convergence to feasability.

    Returns:
        The channel fidelity between the channels specified by the quantum channels corresponding to the Choi matrices
        `choi_1` and `choi_2`.

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

    Examples:
        For two identical channels, we should expect that the channel fidelity should yield a value of
        \(1\).

        ```python exec="1" source="above" result="text"
        import numpy as np
        from toqito.channels import dephasing
        from toqito.channel_metrics import channel_fidelity
        # The Choi matrices of dimension-4 for the dephasing channel
        choi_1 = dephasing(4)
        choi_2 = dephasing(4)
        print(channel_fidelity(choi_1, choi_2))
        ```

        We can also compute the channel fidelity between two different channels. For example, we can
        compute the channel fidelity between the dephasing and depolarizing channels.

        ```python exec="1" source="above" result="text"
        import numpy as np
        from toqito.channels import dephasing, depolarizing
        from toqito.channel_metrics import channel_fidelity
        # The Choi matrices of dimension-4 for the dephasing and depolarizing channels
        choi_1 = dephasing(4)
        choi_2 = depolarizing(4)
        print(channel_fidelity(choi_1, choi_2))
        ```

    """
    if choi_1.shape != choi_2.shape:
        raise ValueError("The Choi matrices provided should be of equal dimension.")

    choi_dim_x, choi_dim_y = choi_1.shape
    if choi_dim_x != choi_dim_y:
        raise ValueError("The Choi matrix provided must be square.")

    choi_dim = choi_dim_x
    dim = int(np.log2(choi_dim))

    lam = cvxpy.Variable(nonneg=True)
    q_var = cvxpy.Variable((choi_dim, choi_dim), complex=True)

    constraints = []
    objective = cvxpy.Maximize(lam)

    constraints.append(cvxpy.bmat([[choi_1, q_var.H], [q_var, choi_2]]) >> 0)

    constraints.append(lam * np.identity(dim) <= cvxpy.real(partial_trace(q_var, [1], [dim, dim])))

    problem = cvxpy.Problem(objective, constraints)

    with warnings.catch_warnings():
        warnings.filterwarnings("ignore", message="Solution may be inaccurate")
        return problem.solve(solver=cvxpy.SCS, eps=eps)