"""The realignment channel."""
from __future__ import annotations
import numpy as np
from toqito.perms import swap
from toqito.channels import partial_transpose
[docs]
def realignment(input_mat: np.ndarray, dim: int | list[int] = None) -> np.ndarray:
r"""
Compute the realignment of a bipartite operator [LAS08]_.
Gives the realignment of the matrix :code:`input_mat`, where it is assumed that the number
of rows and columns of :code:`input_mat` are both perfect squares and both subsystems have
equal dimension. The realignment is defined by mapping the operator :math:`|ij \rangle
\langle kl |` to :math:`|ik \rangle \langle jl |` and extending linearly.
If :code:`input_mat` is non-square, different row and column dimensions can be specified by
putting the row dimensions in the first row of :code:`dim` and the column dimensions in the
second row of :code:`dim`.
Examples
==========
The standard realignment map
Using :code:`toqito`, we can generate the standard realignment map as follows. When viewed as a
map on block matrices, the realignment map takes each block of the original matrix and makes
its vectorization the rows of the realignment matrix. This is illustrated by the following
small example:
>>> from toqito.channels import realignment
>>> import numpy as np
>>> test_input_mat = np.array(
>>> [[1, 2, 3, 4], [5, 6, 7, 8], [9, 10, 11, 12], [13, 14, 15, 16]]
>>> )
>>> realignment(test_input_mat)
[[ 1 2 5 6]
[ 3 4 7 8]
[ 9 10 13 14]
[11 12 15 16]]
References
==========
.. [LAS08] Lupo, Cosmo, Paolo, Aniello, and Scardicchio, Antonello.
"Bipartite quantum systems: on the realignment criterion and beyond."
Journal of Physics A: Mathematical and Theoretical
41.41 (2008): 415301.
https://arxiv.org/abs/0802.2019
:param input_mat: The input matrix.
:param dim: Default has all equal dimensions.
:raises ValueError: If dimension of matrix is invalid.
:return: The realignment map matrix.
"""
eps = np.finfo(float).eps
dim_mat = input_mat.shape
round_dim = np.round(np.sqrt(dim_mat))
if dim is None:
dim = np.transpose(np.array([round_dim]))
if isinstance(dim, list):
dim = np.array(dim)
if isinstance(dim, int):
dim = np.array([int(dim), int(dim_mat[0] / dim)])
if np.abs(dim[1] - np.round(dim[1])) >= 2 * dim_mat[0] * eps:
raise ValueError("InvalidDim:")
dim[1] = np.round(dim[1])
# Dimension if row vector.
if len(dim.shape) == 1:
dim = dim[:].T
dim = np.array([dim, dim])
# Dimension is column vector.
if min(dim.shape) == 1:
dim = dim[:].T[0]
dim = np.array([dim, dim])
dim_x = np.array([[dim[0][1], dim[0][0]], [dim[1][0], dim[1][1]]])
dim_y = np.array([[dim[1][0], dim[0][0]], [dim[0][1], dim[1][1]]])
x_tmp = swap(input_mat, [1, 2], dim, True)
y_tmp = partial_transpose(x_tmp, sys=1, dim=dim_x)
return swap(y_tmp, [1, 2], dim_y, True)