def partial_trace_cvx(rho, sys=None, dim=None):
"""
Perform the partial trace on a :code:`cvxpy` variable.
Adapted from [CVXPtrace]_.
References
==========
.. [CVXPtrace] Partial trace for CVXPY variables
https://github.com/cvxgrp/cvxpy/issues/563
:param rho: A square matrix.
:param sys: Scalar or vector specifying the size of the subsystems.
:param dim: Dimension of the subsystems. If :code:`None`, all dimensions
are assumed to be equal.
:return: The partial trace of matrix :code:`input_mat`.
"""
rho_np = expr_as_np_array(rho)
traced_rho = partial_trace(rho_np, sys, dim)
traced_rho = np_array_as_expr(traced_rho)
return traced_rho
def partial_transpose(
rho: Union[np.ndarray, Variable],
sys: Union[List[int], np.ndarray, int] = 2,
dim: Union[List[int], np.ndarray] = None,
) -> Union[np.ndarray, Expression]:
r"""Compute the partial transpose of a matrix [WikPtrans]_.
The *partial transpose* is defined as
.. math::
\left( \text{T} \otimes \mathbb{I}_{\mathcal{Y}} \right)
\left(X\right)
where :math:`X \in \text{L}(\mathcal{X})` is a linear operator over the
complex Euclidean space :math:`\mathcal{X}` and where :math:`\text{T}` is
the transpose mapping :math:`\text{T} \in \text{T}(\mathcal{X})` defined as
.. math::
\text{T}(X) = X^{\text{T}}
for all :math:`X \in \text{L}(\mathcal{X})`.
By default, the returned matrix is the partial transpose of the matrix
:code:`rho`, where it is assumed that the number of rows and columns of
:code:`rho` are both perfect squares and both subsystems have equal
dimension. The transpose is applied to the second subsystem.
In the case where :code:`sys` amd :code:`dim` are specified, this function
gives the partial transpose of the matrix :code:`rho` where the dimensions
of the (possibly more than 2) subsystems are given by the vector :code:`dim`
and the subsystems to take the partial transpose are given by the scalar or
vector :code:`sys`. If :code:`rho` 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
==========
Consider the following matrix
.. math::
X = \begin{pmatrix}
1 & 2 & 3 & 4 \\
5 & 6 & 7 & 8 \\
9 & 10 & 11 & 12 \\
13 & 14 & 15 & 16
\end{pmatrix}.
Performing the partial transpose on the matrix :math:`X` over the second
subsystem yields the following matrix
.. math::
X_{pt, 2} = \begin{pmatrix}
1 & 5 & 3 & 7 \\
2 & 6 & 4 & 8 \\
9 & 13 & 11 & 15 \\
10 & 14 & 12 & 16
\end{pmatrix}.
By default, in :code:`toqito`, the partial transpose function performs the
transposition on the second subsystem as follows.
>>> from toqito.channels import partial_transpose
>>> import numpy as np
>>> test_input_mat = np.arange(1, 17).reshape(4, 4)
>>> partial_transpose(test_input_mat)
[[ 1 5 3 7]
[ 2 6 4 8]
[ 9 13 11 15]
[10 14 12 16]]
By specifying the :code:`sys = 1` argument, we can perform the partial
transpose over the first subsystem (instead of the default second subsystem
as done above). Performing the partial transpose over the first subsystem
yields the following matrix
.. math::
X_{pt, 1} = \begin{pmatrix}
1 & 2 & 9 & 10 \\
5 & 6 & 13 & 14 \\
3 & 4 & 11 & 12 \\
7 & 8 & 15 & 16
\end{pmatrix}.
>>> from toqito.channels import partial_transpose
>>> 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]]
>>> )
>>> partial_transpose(test_input_mat, 1)
[[ 1 2 9 10]
[ 5 6 13 14]
[ 3 4 11 12]
[ 7 8 15 16]]
References
==========
.. [WikPtrans] Wikipedia: Partial transpose
https://en.wikipedia.org/w/index.php?title=Partial_transpose
:param rho: A matrix.
:param sys: Scalar or vector specifying the size of the subsystems.
:param dim: Dimension of the subsystems. If :code:`None`, all dimensions
are assumed to be equal.
:returns: The partial transpose of matrix :code:`rho`.
"""
# If the input matrix is a CVX variable for an SDP, we convert it to a
# numpy array, perform the partial transpose, and convert it back to a CVX
# variable.
if isinstance(rho, Variable):
rho_np = expr_as_np_array(rho)
transposed_rho = partial_transpose(rho_np, sys, dim)
transposed_rho = np_array_as_expr(transposed_rho)
return transposed_rho
sqrt_rho_dims = np.round(np.sqrt(list(rho.shape)))
if dim is None:
dim = np.array([[sqrt_rho_dims[0], sqrt_rho_dims[0]],
[sqrt_rho_dims[1], sqrt_rho_dims[1]]])
if isinstance(dim, float):
dim = np.array([dim])
if isinstance(dim, list):
dim = np.array(dim)
if isinstance(sys, list):
sys = np.array(sys)
if isinstance(sys, int):
sys = np.array([sys])
num_sys = len(dim)
# Allow the user to enter a single number for dim.
if num_sys == 1:
dim = np.array([dim, list(rho.shape)[0] / dim])
if (np.abs(dim[1] - np.round(dim[1]))[0] >=
2 * list(rho.shape)[0] * np.finfo(float).eps):
raise ValueError("InvalidDim: If `dim` is a scalar, `rho` must be "
"square and `dim` must evenly divide `len(rho)`; "
"please provide the `dim` array containing the "
"dimensions of the subsystems.")
dim[1] = np.round(dim[1])
num_sys = 2
# Allow the user to enter a vector for dim if X is square.
if min(dim.shape) == 1 or len(dim.shape) == 1:
# Force dim to be a row vector.
dim = dim.T.flatten()
dim = np.array([dim, dim])
prod_dim_r = int(np.prod(dim[0, :]))
prod_dim_c = int(np.prod(dim[1, :]))
sub_prod_r = np.prod(dim[0, sys - 1])
sub_prod_c = np.prod(dim[1, sys - 1])
sub_sys_vec_r = prod_dim_r * np.ones(int(sub_prod_r)) / sub_prod_r
sub_sys_vec_c = prod_dim_c * np.ones(int(sub_prod_c)) / sub_prod_c
set_diff = list(set(list(range(1, num_sys + 1))) - set(sys))
perm = sys.tolist()[:]
perm.extend(set_diff)
# Permute the subsystems so that we just have to do the partial transpose
# on the first (potentially larger) subsystem.
rho_permuted = permute_systems(rho, perm, dim)
x_tmp = np.reshape(
rho_permuted,
[
int(sub_sys_vec_r[0]),
int(sub_prod_r),
int(sub_sys_vec_c[0]),
int(sub_prod_c),
],
order="F",
)
y_tmp = np.transpose(x_tmp, [0, 3, 2, 1])
z_tmp = np.reshape(
y_tmp,
[
int(sub_sys_vec_r[0]) * int(sub_prod_c),
int(sub_sys_vec_c[0]) * int(sub_prod_r),
],
order="F",
)
# If z_tmp is a just a 1-D matrix, extract just the row.
if z_tmp.shape[0] == 1:
z_tmp = z_tmp[0]
# Return the subsystems back to their original positions.
dim[[0, 1], sys - 1] = dim[[1, 0], sys - 1]
return permute_systems(z_tmp, perm, dim, False, True)
def partial_trace(
input_mat: Union[np.ndarray, Variable],
sys: Union[int, List[int]] = 2,
dim: Union[int, List[int]] = None,
) -> Union[np.ndarray, Expression]:
r"""
Compute the partial trace of a matrix [WikPtrace]_.
The *partial trace* is defined as
.. math::
\left( \text{Tr} \otimes \mathbb{I}_{\mathcal{Y}} \right)
\left(X \otimes Y \right) = \text{Tr}(X)Y
where :math:`X \in \text{L}(\mathcal{X})` and :math:`Y \in \text{L}(\mathcal{Y})` are linear
operators over complex Euclidean spaces :math:`\mathcal{X}` and :math:`\mathcal{Y}`.
Gives the partial trace of the matrix X, where the dimensions of the (possibly more than 2)
subsystems are given by the vector :code:`dim` and the subsystems to take the trace on are
given by the scalar or vector :code:`sys`.
Examples
==========
Consider the following matrix
.. math::
X = \begin{pmatrix}
1 & 2 & 3 & 4 \\
5 & 6 & 7 & 8 \\
9 & 10 & 11 & 12 \\
13 & 14 & 15 & 16
\end{pmatrix}.
Taking the partial trace over the second subsystem of :math:`X` yields the following matrix
.. math::
X_{pt, 2} = \begin{pmatrix}
7 & 11 \\
23 & 27
\end{pmatrix}
By default, the partial trace function in :code:`toqito` takes the trace of the second
subsystem.
>>> from toqito.channels import partial_trace
>>> 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]]
>>> )
>>> partial_trace(test_input_mat)
[[ 7, 11],
[23, 27]]
By specifying the :code:`sys = 1` argument, we can perform the partial trace over the first
subsystem (instead of the default second subsystem as done above). Performing the partial
trace over the first subsystem yields the following matrix
.. math::
X_{pt, 1} = \begin{pmatrix}
12 & 14 \\
20 & 22
\end{pmatrix}
>>> from toqito.channels import partial_trace
>>> 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]]
>>> )
>>> partial_trace(test_input_mat, 1)
[[12, 14],
[20, 22]]
We can also specify both dimension and system size as :code:`list` arguments. Consider the
following :math:`16`-by-:math:`16` matrix.
>>> from toqito.channels import partial_trace
>>> import numpy as np
>>> test_input_mat = np.arange(1, 257).reshape(16, 16)
>>> test_input_mat
[[ 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16]
[ 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32]
[ 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48]
[ 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64]
[ 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80]
[ 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96]
[ 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112]
[113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128]
[129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144]
[145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160]
[161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176]
[177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192]
[193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208]
[209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224]
[225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240]
[241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256]]
We can take the partial trace on the first and third subsystems and assume that the size of
each of the 4 systems is of dimension 2.
>>> from toqito.channels import partial_trace
>>> import numpy as np
>>> partial_trace(test_input_mat, [1, 3], [2, 2, 2, 2])
[[344, 348, 360, 364],
[408, 412, 424, 428],
[600, 604, 616, 620],
[664, 668, 680, 684]])
References
==========
.. [WikPtrace] Wikipedia: Partial trace
https://en.wikipedia.org/wiki/Partial_trace
:param input_mat: A square matrix.
:param sys: Scalar or vector specifying the size of the subsystems.
:param dim: Dimension of the subsystems. If :code:`None`, all dimensions are assumed to be
equal.
:return: The partial trace of matrix :code:`input_mat`.
"""
# If the input matrix is a CVX variable for an SDP, we convert it to a numpy array,
# perform the partial trace, and convert it back to a CVX variable.
if isinstance(input_mat, Variable):
rho_np = expr_as_np_array(input_mat)
traced_rho = partial_trace(rho_np, sys, dim)
traced_rho = np_array_as_expr(traced_rho)
return traced_rho
if dim is None:
dim = np.array([np.round(np.sqrt(len(input_mat)))])
if isinstance(dim, int):
dim = np.array([dim])
if isinstance(dim, list):
dim = np.array(dim)
num_sys = len(dim)
# Allow the user to enter a single number for dim.
if num_sys == 1:
dim = np.array([dim[0], len(input_mat) / dim[0]])
if np.abs(dim[1] - np.round(dim[1])
) >= 2 * len(input_mat) * np.finfo(float).eps:
raise ValueError(
"Invalid: If `dim` is a scalar, `dim` must evenly "
"divide `len(input_mat)`.")
dim[1] = np.round(dim[1])
num_sys = 2
prod_dim = np.prod(dim)
if isinstance(sys, list):
if len(sys) == 1:
prod_dim_sys = np.prod(dim[sys[0] - 1])
else:
prod_dim_sys = 1
for idx in sys:
prod_dim_sys *= dim[idx - 1]
elif isinstance(sys, int):
prod_dim_sys = np.prod(dim[sys - 1])
else:
raise ValueError("Invalid: The variable `sys` must either be of type "
"int or of a list of ints.")
sub_prod = prod_dim / prod_dim_sys
sub_sys_vec = prod_dim * np.ones(int(sub_prod)) / sub_prod
if isinstance(sys, int):
sys = [sys]
set_diff = list(set(list(range(1, num_sys + 1))) - set(sys))
perm = set_diff
perm.extend(sys)
a_mat = permute_systems(input_mat, perm, dim)
ret_mat = np.reshape(
a_mat,
[
int(sub_sys_vec[0]),
int(sub_prod),
int(sub_sys_vec[0]),
int(sub_prod)
],
order="F",
)
permuted_mat = ret_mat.transpose((1, 3, 0, 2))
permuted_reshaped_mat = np.reshape(
permuted_mat,
[int(sub_prod), int(sub_prod),
int(sub_sys_vec[0]**2)],
order="F",
)
pt_mat = permuted_reshaped_mat[:, :,
list(
range(0, int(sub_sys_vec[0]**2
), int(sub_sys_vec[0] +
1)))]
pt_mat = np.sum(pt_mat, axis=2)
return pt_mat
def test_expr_as_np_array():
"""Ensure return type is numpy object."""
expr = cvxpy.bmat([[1, 2], [3, 4]])
res_mat = expr_as_np_array(expr)
np.testing.assert_equal(isinstance(res_mat, np.ndarray), True)
def test_expr_as_np_array_vector():
"""Ensure return type is numpy object for vector expression."""
cvx_var = cvxpy.Parameter(5)
np.testing.assert_equal(isinstance(expr_as_np_array(cvx_var), np.ndarray), True)
def test_expr_as_np_array_scalar():
"""Ensure return type is numpy object for scalar expression."""
cvx_var = cvxpy.Variable()
np.testing.assert_equal(isinstance(expr_as_np_array(cvx_var), np.ndarray), True)
