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departed-0.3.0
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مشاهده بیشترتوضیحات
Partial trace and partial transpose for matrices with Kronecker product structure.
| ویژگی | مقدار |
|---|---|
| سیستم عامل | - |
| نام فایل | departed-0.3.0 |
| نام | departed |
| نسخه کتابخانه | 0.3.0 |
| نگهدارنده | [] |
| ایمیل نگهدارنده | [] |
| نویسنده | Jan Provaznik |
| ایمیل نویسنده | jan@provaznik.pro |
| آدرس صفحه اصلی | https://provaznik.pro/departed |
| آدرس اینترنتی | https://pypi.org/project/departed/ |
| مجوز | LGPL |
DEPARTED provides partial trace and partial transpose operations for
matrices with a Kronecker-product structure.
Installation
As of release 0.3.0, it is possible to retrieve the latest release of the
package from the PyPi repository by running 'pip install --user departed'
Examples
See the examples directory.
Documentation
- ptrace (matrix, component_dims, component_mask)
Computes the partial trace of a matrix with a Kronecker-product structure.
: matrix
defines the matrix to be partially traced
: component_dims
defines a list of component dimensions
: component_mask
defines a mask specifying whether a component should be
traced out (1, True) or kept (0, False).
Example
Consider a density matrix R of a three-qubit quantum state. The partial
trace over the 1st and 3rd qubit is obtained with
ptrace(R, [ 2, 2, 2 ], [ 1, 0, 1 ]),
where the mask [ 1, 0, 1 ] uses 1 for components that will be traced out.
- ptranspose (matrix, component_dims, component_mask)
Computes the partial transpose of a matrix with a Kronecker-product structure.
: matrix
defines the matrix to be partially transposed
: component_dims
defines a list of component dimensions
: component_mask
defines a mask specifying whether a component should be
transposed (1, True) or left intact (0, False).
Example
Consider a density matrix R of a three-qubit quantum state. The partial
transpose over the 1st and 3rd qubit is obtained with
ptranspose(R, [ 2, 2, 2 ], [ 1, 0, 1 ]),
where the mask [ 1, 0, 1 ] uses 1 for components that will be transposed.
- mask_from_component_list (component_list, mask_width, invert = False)
Constructs a component_mask from a list of component indices.
By default, 1 is set for components listed in index_list and 0 for those
unlisted. These values can be inverted by setting invert = True.
: component_list
defines a list of component indices (starting from 0)
: mask_width
defines the total number of components (mask width)
: invert
defines how component_list is translated into component_mask
Example
Consider a system with 5 components. Suppose the partial operation of
choice is performed over the 2nd and 4th components. The
corresponding mask [ 0, 1, 0, 1, 0 ] can be constructed with
mask_from_component_list([ 1, 3 ], 5),
where we note that the indices start from 0.
Suppose that the partial operation should NOT affect the 3rd and 5th
component. The respective mask [ 1, 1, 0, 1, 0 ] can be constructed with
mask_from_component_list([ 2, 4 ], 5, invert = True),
where setting the optional parameter (invert = True) inverts the mask.
Operating principles of Kronecker product and tensor representation
An understanding of the internal memory layout of matrices with a
Kronecker-product structure is crucial for both the partial trace and partial
transpose operations. In both algorithms, the matrix is first reshaped into a
tensor with the axes of its constituent matrices arranged according to their
direction, with row axes preceding column axes.
In particular, consider a matrix with a Kronecker-product structure
comprising N matrices and its tensor representation. The first N axes of the
tensor are the row axes of its matrix constituents and the last N axes are
their column axes.
This ordering appears counter-intuitive, however, it follows from the
internal memory layout of the underlying numpy.ndarray object.
To illustrate this concept in greater detail, consider a tensor product of
three matrix spaces, where the first matrix space comprises (m x m) matrices,
the second one (n x n) matrices, and finally, the third one (o x o) matrices.
The product space contains (mno x mno) matrices.
Let R be a matrix (with a Kronecker-product structure) from this product
space. Its tensor representation can be obtained through
T = R.reshape([ m, n, o, m, n, o ]),
where the first three axes of T each correspond to the row axes of matrices
from the constituent matrix spaces. The last three axes of T each correspond
to the column axes of these matrices.
For example, should one have three matrices - A, B, and C - and their
Kronecker product, kron(kron(A, B), C), the element
T[:, 0, 2, :, 1, 3 ]
of its tensor representation equals to
A * B[0, 1], * C[2, 3].
Operating principles of partial transpose
Consider an arbitrary bipartite quantum state
rho = sum(i, j, k, l) Q(i, j, k, l) |i><j| |k><l|.
We define the partial transpose over the second system as
pt(rho) = sum(i, j, k, l) Q(i, j, k, l) |i><j| |l><k|
= sum(i, j, k, l) Q(i, j, l, k) |i><j| |k><l|.
If the density operator is encoded in a matrix with a Kronecker-product
structure, the partial transpose no longer amounts to a simple exchange of
axes. It can be realized by determining the affected indices of the matrix
and exchanging their values. Not only is this approach cumbersome, but also
extremely inelegant.
A better alternative is to reshape the matrix into a tensor of appropriate
rank. The reshaping operation is computationally cheap as it only affects the
interpretation of the underlying numpy.ndarray object.
Suppose R is the matrix representation of the state rho. Suppose m and n are
the dimensions of the individual systems comprising the state. With
T = reshape(R, [ m, n, m, n ])
we obtain a rank four tensor T. But alas, the tensor T is not the same as
the tensor Q. The order of their indices is different. We have
T(i, k, j, l)
instead of the theoretical Q(i, j, k, l). Consequently, the partial transpose
corresponds to the exchange of the 2nd and 4th indices
pt(T)(i, k, j, l) = T(i, l, j, k).
We can reconstruct the Kronecker representation by reshaping the tensor back
into the original matrix shape.
This idea can be generalized to an arbitrary number of components. It is
practical to use the standard numpy.transpose procedure and supply the
correct axial permutation. Our ptranspose does exactly that.
Operating principles of partial trace
We build on the same ideas we have explored for partial transpose and then
add a figurative cherry on top: by reordering the tensor, we can perform only
a single trace operation even if multiple components are to be traced out.
Let us begin with a bipartite system again with a density matrix
rho = sum(i, j, k, l) Q(i, j, k, l) |i><j| |k><l|.
We define the partial trace performed over its second component as
rho' = sum(u) sum(i, j, k, l) Q(i, j, k, l) |i><j| <u|k> <l|u>
= sum(i, j, k) Q(i, j, k, k) |i><j|
= sum(i, j) [ sum(k) Q(i, j, k, k) ] |i><j|,
where we have presumed that the individual { |k>, |l> } kets are orthonormal.
If the density operator is encoded in a matrix with a Kronecker-product
structure, the partial trace no longer amounts to a sum over the affected
axes. We have already established that the matrix R can be reshaped into
T(i, k, j, l),
a tensor with indices different from Q(i, j, k, l). This means that if we
wish to obtain the partial trace, we must evaluate
M(i, j) = sum(k) T(i, k, j, k),
which defines the elements M(i, j) of the marginal tensor. This approach can
be extended for multi-partite systems and further optimized. We can permute
the axes to gather the parts to be traced out into a single block and then
compute the trace in one go.
Notes and acknowledgments
This project was inspired by QuTiP (https://github.com/qutip/qutip) and their
essentially identical implementation. This project aims to explain the
underlying principles behind these functions.
نحوه نصب
نصب پکیج whl departed-0.3.0:
pip install departed-0.3.0.whl
نصب پکیج tar.gz departed-0.3.0:
pip install departed-0.3.0.tar.gz