toqito.channel_ops.choi_to_kraus¶

Computes a list of Kraus operators from the Choi matrix.

Module Contents¶

toqito.channel_ops.choi_to_kraus.choi_to_kraus(choi_mat, tol=1e-09, dim=None)[source]¶

Compute a list of Kraus operators from the Choi matrix from [@Rigetti_2022_Forest].

Note that unlike the Choi or natural representation of operators, the Kraus representation is not unique.

If the input channel maps (M_{r,c}) to (M_{x,y}) then dim should be the list [[r,x], [c,y]]. If it maps (M_m) to (M_n), then dim can simply be the vector [m,n].

For completely positive maps the output is a single flat list of numpy arrays since the left and right Kraus maps are the same.

This function has been adapted from [@Rigetti_2022_Forest] and QETLAB [@QETLAB_link].

Examples

Consider taking the Kraus operators of the Choi matrix that characterizes the “swap operator” defined as

[

begin{pmatrix}: 1 & 0 & 0 & 0 \ 0 & 0 & 1 & 0 \ 0 & 1 & 0 & 0 \ 0 & 0 & 0 & 1

end{pmatrix}

]

The corresponding Kraus operators of the swap operator are given as follows,

[

begin{equation} big[

frac{1}{sqrt{2}} begin{pmatrix} 0 & 1 \ -1 & 0 end{pmatrix}, frac{1}{sqrt{2}} begin{pmatrix} 0 & -1 \ 1 & 0 end{pmatrix}

big], big[

frac{1}{sqrt{2}} begin{pmatrix} 0 & 1 \ 1 & 0 end{pmatrix}, frac{1}{sqrt{2}} begin{pmatrix} 0 & 1 \ 1 & 0 end{pmatrix}

big], big[

begin{pmatrix} 1 & 0 \ 0 & 0 end{pmatrix}, begin{pmatrix} 1 & 0 \ 0 & 0 end{pmatrix}

big], big[

begin{pmatrix} 0 & 0 \ 0 & 1 end{pmatrix}, begin{pmatrix} 0 & 0 \ 0 & 1 end{pmatrix}

big] end{equation}

]

This can be verified in |toqito⟩ as follows.

```python exec=”1” source=”above” import numpy as np from toqito.channel_ops import choi_to_kraus choi_mat = np.array([[1, 0, 0, 0], [0, 0, 1, 0], [0, 1, 0, 0], [0, 0, 0, 1]]) kraus_ops = choi_to_kraus(choi_mat) for i, pair in enumerate(kraus_ops):

print(f”nKraus Pair {i+1}:”) for j, op in enumerate(pair):

print(f” Operator {j+1}:n{np.array_str(op, precision=4, suppress_small=True)}”)

```

!!! See Also: [kraus_to_choi][toqito.channel_ops.kraus_to_choi.kraus_to_choi]

Parameters:

choi_mat (numpy.ndarray) – A Choi matrix
tol (float) – optional threshold parameter for eigenvalues/kraus ops to be discarded
dim (int | list[int] | numpy.ndarray | None) – A scalar, vector or matrix containing the input and output dimensions of Choi matrix.

Returns:

List of Kraus operators

Return type:

list[numpy.ndarray] | list[list[numpy.ndarray]]