toqito.channels.choi¶
Generates the Choi channel.
Module Contents¶
- toqito.channels.choi.choi(a_var=1, b_var=1, c_var=0)[source]¶
Produce the Choi channel or one of its generalizations [@Choi_1992_Generalized].
The Choi channel is a positive map on 3-by-3 matrices that is capable of detecting some entanglement that the transpose map is not.
The standard Choi channel defined with a=1, b=1, and c=0 is the Choi matrix of the positive map defined in [@Choi_1992_Generalized]. Many of these maps are capable of detecting PPT entanglement.
Examples
The standard Choi channel is given as
- [
Phi_{1, 1, 0} = begin{pmatrix}
1 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & -1 \ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \ -1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & -1 \ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \ -1 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 1
end{pmatrix}
]
We can generate the Choi channel in |toqito⟩ as follows.
```python exec=”1” source=”above” import numpy as np from toqito.channels import choi
The reduction channel is the map (R) defined by:
- [
R(X) = text{Tr}(X) mathbb{I} - X.
]
The matrix correspond to this is given as
- [
Phi_{0, 1, 1} = begin{pmatrix}
0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & -1 \ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \ -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 \ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \ -1 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0
end{pmatrix}
]
The reduction channel is the Choi channel that arises when a = 0 and when b = c = 1. We can obtain this matrix using |toqito⟩ as follows.
```python exec=”1” source=”above” import numpy as np from toqito.channels import choi
- !!! See Also
[reduction][toqito.channels.reduction.reduction]
- Parameters:
a_var (int) – Default integer for standard Choi map.
b_var (int) – Default integer for standard Choi map.
c_var (int) – Default integer for standard Choi map.
- Returns:
The Choi channel (or one of its generalizations).
- Return type:
numpy.ndarray