toqito.channel_props.choi_rank

Calculates the Choi rank of a channel.

Module Contents

toqito.channel_props.choi_rank.choi_rank(phi)

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.

Parameters:

phi (numpy.ndarray | list[list[numpy.ndarray]]) – Either a Choi matrix or a list of Kraus operators

Returns:

The Choi rank of the provided channel representation.

Return type:

int