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"""Abstract tensor product."""
from sympy.core.add import Add
from sympy.core.expr import Expr
from sympy.core.kind import KindDispatcher
from sympy.core.mul import Mul
from sympy.core.power import Pow
from sympy.core.sympify import sympify
from sympy.matrices.dense import DenseMatrix as Matrix
from sympy.matrices.immutable import ImmutableDenseMatrix as ImmutableMatrix
from sympy.printing.pretty.stringpict import prettyForm
from sympy.utilities.exceptions import sympy_deprecation_warning
from sympy.physics.quantum.dagger import Dagger
from sympy.physics.quantum.kind import (
KetKind, _KetKind,
BraKind, _BraKind,
OperatorKind, _OperatorKind
)
from sympy.physics.quantum.matrixutils import (
numpy_ndarray,
scipy_sparse_matrix,
matrix_tensor_product
)
from sympy.physics.quantum.state import Ket, Bra
from sympy.physics.quantum.trace import Tr
__all__ = [
'TensorProduct',
'tensor_product_simp'
]
#-----------------------------------------------------------------------------
# Tensor product
#-----------------------------------------------------------------------------
_combined_printing = False
def combined_tensor_printing(combined):
"""Set flag controlling whether tensor products of states should be
printed as a combined bra/ket or as an explicit tensor product of different
bra/kets. This is a global setting for all TensorProduct class instances.
Parameters
----------
combine : bool
When true, tensor product states are combined into one ket/bra, and
when false explicit tensor product notation is used between each
ket/bra.
"""
global _combined_printing
_combined_printing = combined
class TensorProduct(Expr):
"""The tensor product of two or more arguments.
For matrices, this uses ``matrix_tensor_product`` to compute the Kronecker
or tensor product matrix. For other objects a symbolic ``TensorProduct``
instance is returned. The tensor product is a non-commutative
multiplication that is used primarily with operators and states in quantum
mechanics.
Currently, the tensor product distinguishes between commutative and
non-commutative arguments. Commutative arguments are assumed to be scalars
and are pulled out in front of the ``TensorProduct``. Non-commutative
arguments remain in the resulting ``TensorProduct``.
Parameters
==========
args : tuple
A sequence of the objects to take the tensor product of.
Examples
========
Start with a simple tensor product of SymPy matrices::
>>> from sympy import Matrix
>>> from sympy.physics.quantum import TensorProduct
>>> m1 = Matrix([[1,2],[3,4]])
>>> m2 = Matrix([[1,0],[0,1]])
>>> TensorProduct(m1, m2)
Matrix([
[1, 0, 2, 0],
[0, 1, 0, 2],
[3, 0, 4, 0],
[0, 3, 0, 4]])
>>> TensorProduct(m2, m1)
Matrix([
[1, 2, 0, 0],
[3, 4, 0, 0],
[0, 0, 1, 2],
[0, 0, 3, 4]])
We can also construct tensor products of non-commutative symbols:
>>> from sympy import Symbol
>>> A = Symbol('A',commutative=False)
>>> B = Symbol('B',commutative=False)
>>> tp = TensorProduct(A, B)
>>> tp
AxB
We can take the dagger of a tensor product (note the order does NOT reverse
like the dagger of a normal product):
>>> from sympy.physics.quantum import Dagger
>>> Dagger(tp)
Dagger(A)xDagger(B)
Expand can be used to distribute a tensor product across addition:
>>> C = Symbol('C',commutative=False)
>>> tp = TensorProduct(A+B,C)
>>> tp
(A + B)xC
>>> tp.expand(tensorproduct=True)
AxC + BxC
"""
is_commutative = False
_kind_dispatcher = KindDispatcher("TensorProduct_kind_dispatcher", commutative=True)
@property
def kind(self):
"""Calculate the kind of a tensor product by looking at its children."""
arg_kinds = (a.kind for a in self.args)
return self._kind_dispatcher(*arg_kinds)
def __new__(cls, *args):
if isinstance(args[0], (Matrix, ImmutableMatrix, numpy_ndarray,
scipy_sparse_matrix)):
return matrix_tensor_product(*args)
c_part, new_args = cls.flatten(sympify(args))
c_part = Mul(*c_part)
if len(new_args) == 0:
return c_part
elif len(new_args) == 1:
return c_part * new_args[0]
else:
tp = Expr.__new__(cls, *new_args)
return c_part * tp
@classmethod
def flatten(cls, args):
# TODO: disallow nested TensorProducts.
c_part = []
nc_parts = []
for arg in args:
cp, ncp = arg.args_cnc()
c_part.extend(list(cp))
nc_parts.append(Mul._from_args(ncp))
return c_part, nc_parts
def _eval_adjoint(self):
return TensorProduct(*[Dagger(i) for i in self.args])
def _eval_rewrite(self, rule, args, **hints):
return TensorProduct(*args).expand(tensorproduct=True)
def _sympystr(self, printer, *args):
length = len(self.args)
s = ''
for i in range(length):
if isinstance(self.args[i], (Add, Pow, Mul)):
s = s + '('
s = s + printer._print(self.args[i])
if isinstance(self.args[i], (Add, Pow, Mul)):
s = s + ')'
if i != length - 1:
s = s + 'x'
return s
def _pretty(self, printer, *args):
if (_combined_printing and
(all(isinstance(arg, Ket) for arg in self.args) or
all(isinstance(arg, Bra) for arg in self.args))):
length = len(self.args)
pform = printer._print('', *args)
for i in range(length):
next_pform = printer._print('', *args)
length_i = len(self.args[i].args)
for j in range(length_i):
part_pform = printer._print(self.args[i].args[j], *args)
next_pform = prettyForm(*next_pform.right(part_pform))
if j != length_i - 1:
next_pform = prettyForm(*next_pform.right(', '))
if len(self.args[i].args) > 1:
next_pform = prettyForm(
*next_pform.parens(left='{', right='}'))
pform = prettyForm(*pform.right(next_pform))
if i != length - 1:
pform = prettyForm(*pform.right(',' + ' '))
pform = prettyForm(*pform.left(self.args[0].lbracket))
pform = prettyForm(*pform.right(self.args[0].rbracket))
return pform
length = len(self.args)
pform = printer._print('', *args)
for i in range(length):
next_pform = printer._print(self.args[i], *args)
if isinstance(self.args[i], (Add, Mul)):
next_pform = prettyForm(
*next_pform.parens(left='(', right=')')
)
pform = prettyForm(*pform.right(next_pform))
if i != length - 1:
if printer._use_unicode:
pform = prettyForm(*pform.right('\N{N-ARY CIRCLED TIMES OPERATOR}' + ' '))
else:
pform = prettyForm(*pform.right('x' + ' '))
return pform
def _latex(self, printer, *args):
if (_combined_printing and
(all(isinstance(arg, Ket) for arg in self.args) or
all(isinstance(arg, Bra) for arg in self.args))):
def _label_wrap(label, nlabels):
return label if nlabels == 1 else r"\left\{%s\right\}" % label
s = r", ".join([_label_wrap(arg._print_label_latex(printer, *args),
len(arg.args)) for arg in self.args])
return r"{%s%s%s}" % (self.args[0].lbracket_latex, s,
self.args[0].rbracket_latex)
length = len(self.args)
s = ''
for i in range(length):
if isinstance(self.args[i], (Add, Mul)):
s = s + '\\left('
# The extra {} brackets are needed to get matplotlib's latex
# rendered to render this properly.
s = s + '{' + printer._print(self.args[i], *args) + '}'
if isinstance(self.args[i], (Add, Mul)):
s = s + '\\right)'
if i != length - 1:
s = s + '\\otimes '
return s
def doit(self, **hints):
return TensorProduct(*[item.doit(**hints) for item in self.args])
def _eval_expand_tensorproduct(self, **hints):
"""Distribute TensorProducts across addition."""
args = self.args
add_args = []
for i in range(len(args)):
if isinstance(args[i], Add):
for aa in args[i].args:
tp = TensorProduct(*args[:i] + (aa,) + args[i + 1:])
c_part, nc_part = tp.args_cnc()
# Check for TensorProduct object: is the one object in nc_part, if any:
# (Note: any other object type to be expanded must be added here)
if len(nc_part) == 1 and isinstance(nc_part[0], TensorProduct):
nc_part = (nc_part[0]._eval_expand_tensorproduct(), )
add_args.append(Mul(*c_part)*Mul(*nc_part))
break
if add_args:
return Add(*add_args)
else:
return self
def _eval_trace(self, **kwargs):
indices = kwargs.get('indices', None)
exp = self
if indices is None or len(indices) == 0:
return Mul(*[Tr(arg).doit() for arg in exp.args])
else:
return Mul(*[Tr(value).doit() if idx in indices else value
for idx, value in enumerate(exp.args)])
def tensor_product_simp_Mul(e):
"""Simplify a Mul with tensor products.
.. deprecated:: 1.14.
The transformations applied by this function are not done automatically
when tensor products are combined.
Originally, the main use of this function is to simplify a ``Mul`` of
``TensorProduct``s to a ``TensorProduct`` of ``Muls``.
"""
sympy_deprecation_warning(
"""
tensor_product_simp_Mul has been deprecated. The transformations
performed by this function are now done automatically when
tensor products are multiplied.
""",
deprecated_since_version="1.14",
active_deprecations_target='deprecated-tensorproduct-simp'
)
return e
def tensor_product_simp_Pow(e):
"""Evaluates ``Pow`` expressions whose base is ``TensorProduct``
.. deprecated:: 1.14.
The transformations applied by this function are not done automatically
when tensor products are combined.
"""
sympy_deprecation_warning(
"""
tensor_product_simp_Pow has been deprecated. The transformations
performed by this function are now done automatically when
tensor products are exponentiated.
""",
deprecated_since_version="1.14",
active_deprecations_target='deprecated-tensorproduct-simp'
)
return e
def tensor_product_simp(e, **hints):
"""Try to simplify and combine tensor products.
.. deprecated:: 1.14.
The transformations applied by this function are not done automatically
when tensor products are combined.
Originally, this function tried to pull expressions inside of ``TensorProducts``.
It only worked for relatively simple cases where the products have
only scalars, raw ``TensorProducts``, not ``Add``, ``Pow``, ``Commutators``
of ``TensorProducts``.
"""
sympy_deprecation_warning(
"""
tensor_product_simp has been deprecated. The transformations
performed by this function are now done automatically when
tensor products are combined.
""",
deprecated_since_version="1.14",
active_deprecations_target='deprecated-tensorproduct-simp'
)
return e
@TensorProduct._kind_dispatcher.register(_OperatorKind, _OperatorKind)
def find_op_kind(e1, e2):
return OperatorKind
@TensorProduct._kind_dispatcher.register(_KetKind, _KetKind)
def find_ket_kind(e1, e2):
return KetKind
@TensorProduct._kind_dispatcher.register(_BraKind, _BraKind)
def find_bra_kind(e1, e2):
return BraKind
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