channel_metrics.channel_fidelity

Computes the channel fidelity between two quantum channels.

Functions

channel_fidelity(choi_1, choi_2[, eps])

Compute the channel fidelity between two quantum channels [1].

Module Contents

channel_metrics.channel_fidelity.channel_fidelity(choi_1, choi_2, eps=1e-07)

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],

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

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 \(1\).

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)
channel_fidelity(choi_1, choi_2)

np.float64(0.9999999999309465)

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)
channel_fidelity(choi_1, choi_2)

np.float64(0.5003013055043918)

References

Raises:

  • ValueError – If matrices are not of equal dimension.

  • ValueError – If matrices are not square.

Parameters:

  • choi_1 (numpy.ndarray) – The Choi matrix of the first quantum channel.

  • choi_2 (numpy.ndarray) – The Choi matrix of the second quantum channel.

  • eps (float) – 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.

Return type:

float