toqito.channel_ops.choi_to_kraus
================================

.. py:module:: toqito.channel_ops.choi_to_kraus

.. autoapi-nested-parse::

   Computes a list of Kraus operators from the Choi matrix.





Module Contents
---------------

.. py:function:: choi_to_kraus(choi_mat, tol = 1e-09, dim = None)

   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].

   .. rubric:: 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]

   :param choi_mat: A Choi matrix
   :param tol: optional threshold parameter for eigenvalues/kraus ops to be discarded
   :param dim: A scalar, vector or matrix containing the input and output dimensions of Choi matrix.

   :returns: List of Kraus operators