# Source code for qutip.tensor

```
# This file is part of QuTiP: Quantum Toolbox in Python.
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# Copyright (c) 2011 and later, Paul D. Nation and Robert J. Johansson.
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###############################################################################
"""
Module for the creation of composite quantum objects via the tensor product.
"""
__all__ = [
'tensor', 'super_tensor', 'composite', 'tensor_swap', 'tensor_contract'
]
import numpy as np
import scipy.sparse as sp
from qutip.cy.spmath import zcsr_kron
from qutip.qobj import Qobj
from qutip.permute import reshuffle
from qutip.superoperator import operator_to_vector
from qutip.dimensions import (
flatten, enumerate_flat, unflatten, deep_remove,
dims_to_tensor_shape, dims_idxs_to_tensor_idxs
)
import qutip.settings
import qutip.superop_reps # Avoid circular dependency here.
[docs]def tensor(*args):
"""Calculates the tensor product of input operators.
Parameters
----------
args : array_like
``list`` or ``array`` of quantum objects for tensor product.
Returns
-------
obj : qobj
A composite quantum object.
Examples
--------
>>> tensor([sigmax(), sigmax()])
Quantum object: dims = [[2, 2], [2, 2]], \
shape = [4, 4], type = oper, isHerm = True
Qobj data =
[[ 0.+0.j 0.+0.j 0.+0.j 1.+0.j]
[ 0.+0.j 0.+0.j 1.+0.j 0.+0.j]
[ 0.+0.j 1.+0.j 0.+0.j 0.+0.j]
[ 1.+0.j 0.+0.j 0.+0.j 0.+0.j]]
"""
if not args:
raise TypeError("Requires at least one input argument")
if len(args) == 1 and isinstance(args[0], (list, np.ndarray)):
# this is the case when tensor is called on the form:
# tensor([q1, q2, q3, ...])
qlist = args[0]
elif len(args) == 1 and isinstance(args[0], Qobj):
# tensor is called with a single Qobj as an argument, do nothing
return args[0]
else:
# this is the case when tensor is called on the form:
# tensor(q1, q2, q3, ...)
qlist = args
if not all([isinstance(q, Qobj) for q in qlist]):
# raise error if one of the inputs is not a quantum object
raise TypeError("One of inputs is not a quantum object")
out = Qobj()
if qlist[0].issuper:
out.superrep = qlist[0].superrep
if not all([q.superrep == out.superrep for q in qlist]):
raise TypeError("In tensor products of superroperators, all must" +
"have the same representation")
out.isherm = True
for n, q in enumerate(qlist):
if n == 0:
out.data = q.data
out.dims = q.dims
else:
out.data = zcsr_kron(out.data, q.data)
out.dims = [out.dims[0] + q.dims[0], out.dims[1] + q.dims[1]]
out.isherm = out.isherm and q.isherm
if not out.isherm:
out._isherm = None
return out.tidyup() if qutip.settings.auto_tidyup else out
[docs]def super_tensor(*args):
"""Calculates the tensor product of input superoperators, by tensoring
together the underlying Hilbert spaces on which each vectorized operator
acts.
Parameters
----------
args : array_like
``list`` or ``array`` of quantum objects with ``type="super"``.
Returns
-------
obj : qobj
A composite quantum object.
"""
if isinstance(args[0], list):
args = args[0]
# Check if we're tensoring vectors or superoperators.
if all(arg.issuper for arg in args):
if not all(arg.superrep == "super" for arg in args):
raise TypeError(
"super_tensor on type='super' is only implemented for "
"superrep='super'."
)
# Reshuffle the superoperators.
shuffled_ops = list(map(reshuffle, args))
# Tensor the result.
shuffled_tensor = tensor(shuffled_ops)
# Unshuffle and return.
out = reshuffle(shuffled_tensor)
out.superrep = args[0].superrep
return out
elif all(arg.isoperket for arg in args):
# Reshuffle the superoperators.
shuffled_ops = list(map(reshuffle, args))
# Tensor the result.
shuffled_tensor = tensor(shuffled_ops)
# Unshuffle and return.
out = reshuffle(shuffled_tensor)
return out
elif all(arg.isoperbra for arg in args):
return super_tensor(*(arg.dag() for arg in args)).dag()
else:
raise TypeError(
"All arguments must be the same type, "
"either super, operator-ket or operator-bra."
)
def _isoperlike(q):
return q.isoper or q.issuper
def _isketlike(q):
return q.isket or q.isoperket
def _isbralike(q):
return q.isbra or q.isoperbra
[docs]def composite(*args):
"""
Given two or more operators, kets or bras, returns the Qobj
corresponding to a composite system over each argument.
For ordinary operators and vectors, this is the tensor product,
while for superoperators and vectorized operators, this is
the column-reshuffled tensor product.
If a mix of Qobjs supported on Hilbert and Liouville spaces
are passed in, the former are promoted. Ordinary operators
are assumed to be unitaries, and are promoted using ``to_super``,
while kets and bras are promoted by taking their projectors and
using ``operator_to_vector(ket2dm(arg))``.
"""
# First step will be to ensure everything is a Qobj at all.
if not all(isinstance(arg, Qobj) for arg in args):
raise TypeError("All arguments must be Qobjs.")
# Next, figure out if we have something oper-like (isoper or issuper),
# or something ket-like (isket or isoperket). Bra-like we'll deal with
# by turning things into ket-likes and back.
if all(map(_isoperlike, args)):
# OK, we have oper/supers.
if any(arg.issuper for arg in args):
# Note that to_super does nothing to things
# that are already type=super, while it will
# promote unitaries to superunitaries.
return super_tensor(*map(qutip.superop_reps.to_super, args))
else:
# Everything's just an oper, so ordinary tensor products work.
return tensor(*args)
elif all(map(_isketlike, args)):
# Ket-likes.
if any(arg.isoperket for arg in args):
# We have a vectorized operator, we we may need to promote
# something.
return super_tensor(*(
arg if arg.isoperket
else operator_to_vector(qutip.states.ket2dm(arg))
for arg in args
))
else:
# Everything's ordinary, so we can use the tensor product here.
return tensor(*args)
elif all(map(_isbralike, args)):
# Turn into ket-likes and recurse.
return composite(*(arg.dag() for arg in args)).dag()
else:
raise TypeError("Unsupported Qobj types [{}].".format(
", ".join(arg.type for arg in args)
))
def _tensor_contract_single(arr, i, j):
"""
Contracts a dense tensor along a single index pair.
"""
if arr.shape[i] != arr.shape[j]:
raise ValueError("Cannot contract over indices of different length.")
idxs = np.arange(arr.shape[i])
sl = tuple(slice(None, None, None)
if idx not in (i, j) else idxs for idx in range(arr.ndim))
contract_at = i if j == i + 1 else 0
return np.sum(arr[sl], axis=contract_at)
def _tensor_contract_dense(arr, *pairs):
"""
Contracts a dense tensor along one or more index pairs,
keeping track of how the indices are relabeled by the removal
of other indices.
"""
axis_idxs = list(range(arr.ndim))
for pair in pairs:
# axis_idxs.index effectively evaluates the mapping from
# original index labels to the labels after contraction.
arr = _tensor_contract_single(arr, *map(axis_idxs.index, pair))
list(map(axis_idxs.remove, pair))
return arr
def tensor_swap(q_oper, *pairs):
"""Transposes one or more pairs of indices of a Qobj.
Note that this uses dense representations and thus
should *not* be used for very large Qobjs.
Parameters
----------
pairs : tuple
One or more tuples ``(i, j)`` indicating that the
``i`` and ``j`` dimensions of the original qobj
should be swapped.
Returns
-------
sqobj : Qobj
The original Qobj with all named index pairs swapped with each other
"""
dims = q_oper.dims
tensor_pairs = dims_idxs_to_tensor_idxs(dims, pairs)
data = q_oper.data.toarray()
# Reshape into tensor indices
data = data.reshape(dims_to_tensor_shape(dims))
# Now permute the dims list so we know how to get back.
flat_dims = flatten(dims)
perm = list(range(len(flat_dims)))
for i, j in pairs:
flat_dims[i], flat_dims[j] = flat_dims[j], flat_dims[i]
for i, j in tensor_pairs:
perm[i], perm[j] = perm[j], perm[i]
dims = unflatten(flat_dims, enumerate_flat(dims))
# Next, permute the actual indices of the dense tensor.
data = data.transpose(perm)
# Reshape back, using the left and right of dims.
data = data.reshape(list(map(np.prod, dims)))
return Qobj(inpt=data, dims=dims, superrep=q_oper.superrep)
[docs]def tensor_contract(qobj, *pairs):
"""Contracts a qobj along one or more index pairs.
Note that this uses dense representations and thus
should *not* be used for very large Qobjs.
Parameters
----------
pairs : tuple
One or more tuples ``(i, j)`` indicating that the
``i`` and ``j`` dimensions of the original qobj
should be contracted.
Returns
-------
cqobj : Qobj
The original Qobj with all named index pairs contracted
away.
"""
# Record and label the original dims.
dims = qobj.dims
dims_idxs = enumerate_flat(dims)
tensor_dims = dims_to_tensor_shape(dims)
# Convert to dense first, since sparse won't support the reshaping we need.
qtens = qobj.data.toarray()
# Reshape by the flattened dims.
qtens = qtens.reshape(tensor_dims)
# Contract out the indices from the flattened object.
# Note that we need to feed pairs through dims_idxs_to_tensor_idxs
# to ensure that we are contracting the right indices.
qtens = _tensor_contract_dense(qtens, *dims_idxs_to_tensor_idxs(dims, pairs))
# Remove the contracted indexes from dims so we know how to
# reshape back.
# This concerns dims, and not the tensor indices, so we need
# to make sure to use the original dims indices and not the ones
# generated by dims_to_* functions.
contracted_idxs = deep_remove(dims_idxs, *flatten(list(map(list, pairs))))
contracted_dims = unflatten(flatten(dims), contracted_idxs)
# We don't need to check for tensor idxs versus dims idxs here,
# as column- versus row-stacking will never move an index for the
# vectorized operator spaces all the way from the left to the right.
l_mtx_dims, r_mtx_dims = map(np.product, map(flatten, contracted_dims))
# Reshape back into a 2D matrix.
qmtx = qtens.reshape((l_mtx_dims, r_mtx_dims))
# Return back as a qobj.
return Qobj(qmtx, dims=contracted_dims, superrep=qobj.superrep)
import qutip.states
```