def partial_trace(expr, dims: Tuple[int], axis: Optional[int] = 0):
"""
Assumes :math:`\\texttt{expr} = X_1 \\otimes \\cdots \\otimes X_n` is a 2D Kronecker
product composed of :math:`n = \\texttt{len(dims)}` implicit subsystems.
Letting :math:`k = \\texttt{axis}`, the returned expression represents
the *partial trace* of :math:`\\texttt{expr}` along its :math:`k^{\\text{th}}` implicit subsystem:
.. math::
\\text{tr}(X_k) (X_1 \\otimes \\cdots \\otimes X_{k-1} \\otimes X_{k+1} \\otimes \\cdots \\otimes X_n).
Parameters
----------
expr : :class:`~cvxpy.expressions.expression.Expression`
The 2D expression to take the partial trace of.
dims : tuple of ints.
A tuple of integers encoding the dimensions of each subsystem.
axis : int
The index of the subsystem to be traced out
from the tensor product that defines expr.
"""
expr = Atom.cast_to_const(expr)
if expr.ndim < 2 or expr.shape[0] != expr.shape[1]:
raise ValueError("Only supports square matrices.")
if axis < 0 or axis >= len(dims):
raise ValueError(
f"Invalid axis argument, should be between 0 and {len(dims)}, got {axis}."
)
if expr.shape[0] != np.prod(dims):
raise ValueError(
"Dimension of system doesn't correspond to dimension of subsystems."
)
return sum([_term(expr, j, dims, axis) for j in range(dims[axis])])
def partial_transpose(expr, dims: Tuple[int], axis: Optional[int] = 0):
"""Partial transpose of a matrix.
Assumes :math:`expr = X1 \\odots ... \\odots Xn` is a 2D tensor product
composed implicitly of n subsystems. Here :math:`\\odots` denotes the Kronecker product,
and axis=k is the index of the subsystem to be transposed
from the tensor product that defines expr.
Returns :math:`X1 \\odots ... \\odots Xk^T \\odots ... \\odots Xn`
Parameters
----------
expr : :class:`~cvxpy.expressions.expression.Expression`
The 2D expression to take the partial transpose of.
dims : tuple of ints.
A tuple of integers encoding the dimensions of each subsystem.
axis : int
The index of the subsystem to be transposed
from the tensor product that defines expr.
"""
expr = Atom.cast_to_const(expr)
if expr.ndim < 2 or expr.shape[0] != expr.shape[1]:
raise ValueError("Only supports square matrices.")
if axis < 0 or axis >= len(dims):
raise ValueError(
f"Invalid axis argument, should be between 0 and {len(dims)}, got {axis}."
)
if expr.shape[0] != np.prod(dims):
raise ValueError(
"Dimension of system doesn't correspond to dimension of subsystems."
)
return sum([
_term(expr, i, j, dims, axis) for i in range(dims[axis])
for j in range(dims[axis])
])
def __init__(self, y, parent):
"""
Parameters
----------
y : cvxpy.expressions.expression.Expression
Must satisfy ``y.is_affine() == True``.
parent : cvxpy.transforms.suppfunc.SuppFunc
The object containing data for the convex set associated with this atom.
"""
self.id = lu.get_id()
self.args = [Atom.cast_to_const(y)]
self._parent = parent
self._eta = None # store for debugging purposes
self._shape = tuple()
self.validate_arguments()
def __init__(self, A, X):
# NB: It is important that the exponent is an attribute, not
# an argument. This prevents parametrized exponents from being replaced
# with their logs in Dgp2Dcp.
self.A = Atom.cast_to_const(A)
super(gmatmul, self).__init__(X)
def __init__(self, A, X):
self.A = Atom.cast_to_const(A)
super(gmatmul, self).__init__(X)