toqito.channels.amplitude_damping

Generates the (generalized) amplitude damping channel.

Module Contents

toqito.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 [@Khatri_2020_Information].

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

[

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}, \

]

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:

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

Parameters:

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

• channel. (of the)

• 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 | list[numpy.ndarray]