toqito.channel_props.is_completely_positive

Determines if a channel is completely positive.

Module Contents

toqito.channel_props.is_completely_positive.is_completely_positive(phi, rtol=1e-05, atol=1e-08)

Determine whether the given channel is completely positive.

(Section: Linear Maps Of Square Operators from [@Watrous_2018_TQI]).

A map (Phi in text{T} left(mathcal{X}, mathcal{Y} right)) is completely positive if it holds that

[

Phi otimes mathbb{I}_{text{L}(mathcal{Z})}

]

is a positive map for every complex Euclidean space (mathcal{Z}).

Alternatively, a channel is completely positive if the corresponding Choi matrix of the channel is both Hermitian-preserving and positive semidefinite.

Examples

We can specify the input as a list of Kraus operators. Consider the map (Phi) defined as

[

Phi(X) = X - U X U^*

]

where

[

U = frac{1}{sqrt{2}} begin{pmatrix}

1 & 1 \ -1 & 1

end{pmatrix}.

]

This map is not completely positive, as we can verify as follows.

```python exec=”1” source=”above” import numpy as np from toqito.channel_props import is_completely_positive

unitary_mat = np.array([[1, 1], [-1, 1]]) / np.sqrt(2) kraus_ops = [[np.identity(2), np.identity(2)], [unitary_mat, -unitary_mat]]

print(is_completely_positive(kraus_ops)) ```

We can also specify the input as a Choi matrix. For instance, consider the Choi matrix corresponding to the (2)-dimensional completely depolarizing channel

[

Omega = frac{1}{2} begin{pmatrix}

1 & 0 & 0 & 0 \ 0 & 1 & 0 & 0 \ 0 & 0 & 1 & 0 \ 0 & 0 & 0 & 1

end{pmatrix}.

]

We may verify that this channel is completely positive

```python exec=”1” source=”above” from toqito.channels import depolarizing from toqito.channel_props import is_completely_positive

print(is_completely_positive(depolarizing(2))) ```

Parameters:

• phi (numpy.ndarray | list[list[numpy.ndarray]]) – The channel provided as either a Choi matrix or a list of Kraus operators.

• rtol (float) – The relative tolerance parameter (default 1e-05).

• atol (float) – The absolute tolerance parameter (default 1e-08).

Returns:

True if the channel is completely positive, and False otherwise.

Return type:

bool