"""Calculates the Choi rank of a channel."""
import numpy as np
from toqito.channel_ops import kraus_to_choi
[docs]
def choi_rank(phi: np.ndarray | list[list[np.ndarray]]) -> int:
r"""Calculate the rank of the Choi representation of a quantum channel.
(Section 2.2: Quantum Channels from [@Watrous_2018_TQI]).
Examples:
The transpose map can be written either in Choi representation (as a
SWAP operator) or in Kraus representation. If we choose the latter, it
will be given by the following matrices:
\[
\begin{equation}
\frac{1}{\sqrt{2}}
\begin{pmatrix}
0 & i \\ -i & 0
\end{pmatrix}, \quad
\frac{1}{\sqrt{2}}
\begin{pmatrix}
0 & 1 \\
1 & 0
\end{pmatrix}, \quad
\begin{pmatrix}
1 & 0 \\
0 & 0
\end{pmatrix}, \quad
\begin{pmatrix}
0 & 0 \\
0 & 1
\end{pmatrix}.
\end{equation}
\]
and can be generated in `|toqito⟩` with the following list:
```python exec="1" source="above"
import numpy as np
from toqito.channel_props import choi_rank
kraus_1 = np.array([[1, 0], [0, 0]])
kraus_2 = np.array([[1, 0], [0, 0]]).conj().T
kraus_3 = np.array([[0, 1], [0, 0]])
kraus_4 = np.array([[0, 1], [0, 0]]).conj().T
kraus_5 = np.array([[0, 0], [1, 0]])
kraus_6 = np.array([[0, 0], [1, 0]]).conj().T
kraus_7 = np.array([[0, 0], [0, 1]])
kraus_8 = np.array([[0, 0], [0, 1]]).conj().T
kraus_ops = [[kraus_1, kraus_2], [kraus_3, kraus_4],[kraus_5, kraus_6],[kraus_7, kraus_8]]
print(choi_rank(kraus_ops))
```
We can the verify the associated Choi representation (the SWAP gate)
gets the same Choi rank:
```python exec="1" source="above"
import numpy as np
from toqito.channel_props import choi_rank
choi_matrix = np.array([[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]])
print(choi_rank(choi_matrix))
```
Raises:
ValueError: If matrix is not Choi.
Args:
phi: Either a Choi matrix or a list of Kraus operators
Returns:
The Choi rank of the provided channel representation.
"""
if isinstance(phi, list):
phi = kraus_to_choi(phi)
elif not isinstance(phi, np.ndarray):
raise ValueError("Not a valid Choi matrix.")
return np.linalg.matrix_rank(phi)