Source code for toqito.channel_metrics.channel_fidelity

"""Computes the channel fidelity between two quantum channels."""

import cvxpy
import numpy as np

from toqito.matrix_ops import partial_trace




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 [@Katariya_2021_Geometric].

    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 [@Katariya_2021_Geometric],

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

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

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

        ```python exec="1" source="above"
        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"
        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))
        ```

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

    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`.

    """
    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)

    return problem.solve(solver=cvxpy.SCS, eps=eps)