"""Compute the Choi matrix of a list of Kraus operators."""
import numpy as np
from toqito.states import max_entangled
from toqito.channel_ops import partial_channel
[docs]
def kraus_to_choi(kraus_ops: list[list[np.ndarray]], sys: int = 2) -> np.ndarray:
r"""
Compute the Choi matrix of a list of Kraus operators [WatKraus18]_.
The Choi matrix of the list of Kraus operators, :code:`kraus_ops`. The default convention is
that the Choi matrix is the result of applying the map to the second subsystem of the
standard maximally entangled (unnormalized) state. The Kraus operators are expected to be
input as a list of numpy arrays.
This function was adapted from the QETLAB package.
Examples
==========
The transpose map:
The Choi matrix of the transpose map is the swap operator.
>>> import numpy as np
>>> from toqito.channel_ops import kraus_to_choi
>>> 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],
>>> ]
>>> kraus_to_choi(kraus_ops)
[[1. 0. 0. 0.]
[0. 0. 1. 0.]
[0. 1. 0. 0.]
[0. 0. 0. 1.]]
See Also
========
choi_to_kraus
References
==========
.. [WatKraus18] Watrous, John.
"The theory of quantum information."
Section: "Kraus representations".
Cambridge University Press, 2018.
:param kraus_ops: A list of Kraus operators.
:param sys: The dimension of the system (default is 2).
:return: The corresponding Choi matrix of the provided Kraus operators.
"""
dim_op_1 = kraus_ops[0][0].shape[0]
dim_op_2 = kraus_ops[0][0].shape[1]
choi_mat = partial_channel(
max_entangled(dim_op_1, False, False) * max_entangled(dim_op_2, False, False).conj().T,
kraus_ops,
sys,
np.array([[dim_op_1, dim_op_1], [dim_op_2, dim_op_2]]),
)
return choi_mat