channels.amplitude_damping

Generates the (generalized) amplitude damping channel.

Functions

amplitude_damping([input_mat, gamma, prob])

Apply the generalized amplitude damping channel to a quantum state.

Module Contents

channels.amplitude_damping.amplitude_damping(input_mat=None, gamma=0, prob=1)

Apply the generalized amplitude damping channel to a quantum state.

The generalized amplitude damping channel is a quantum channel that models energy dissipation in a quantum system, where the system can lose energy to its environment with a certain probability. This channel is defined by two parameters: gamma (the damping rate) and prob (the probability of energy loss).

To also include standard implementation of amplitude damping, we have set prob = 1 as the default implementation.

Note

This channel is defined for qubit systems in the standard literature [1].

The Kraus operators for the generalized amplitude damping channel are given by:

\[\begin{split}K_0 = \sqrt{p} \begin{pmatrix} 1 & 0 \\ 0 & \sqrt{1 - \gamma} \end{pmatrix}, \\ K_1 = \sqrt{p} \begin{pmatrix} 0 & \sqrt{\gamma} \\ 0 & 0 \end{pmatrix}, \\ K_2 = \sqrt{1 - p} \begin{pmatrix} \sqrt{1 - \gamma} & 0 \\ 0 & 1 \end{pmatrix}, \\ K_3 = \sqrt{1 - p} \begin{pmatrix} 0 & 0 \\ \sqrt{\gamma} & 0 \end{pmatrix}, \\\end{split}\]

These operators describe the evolution of a quantum state under the generalized amplitude damping process.

Examples

Apply the generalized amplitude damping channel to a qubit state:

import numpy as np
from toqito.channels import amplitude_damping

rho = np.array([[1, 0], [0, 0]])  # |0><0|
result = amplitude_damping(rho, gamma=0.1, prob=0.5)

print(result)

[[0.95+0.j 0.  +0.j]
 [0.  +0.j 0.05+0.j]]

References

[1]

Sumeet Khatri, Kunal Sharma, and Mark M. Wilde. Information-theoretic aspects of the generalized amplitude-damping channel. Phys. Rev. A, 102:012401, Jul 2020. URL: https://link.aps.org/doi/10.1103/PhysRevA.102.012401, doi:10.1103/PhysRevA.102.012401.

Parameters:

  • input_mat (numpy.ndarray | None) – The input matrix to which the channel is applied. If None, the function returns the Kraus operators of the channel.

  • gamma (float) – The damping rate, a float between 0 and 1. Represents the probability of energy dissipation.

  • prob (float) – The probability of energy loss, a float between 0 and 1.

Returns:

The evolved quantum state after applying the generalized amplitude damping channel. If input_mat is None, it returns the list of Kraus operators.

Return type:

numpy.ndarray