departed

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 its 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. Example To illustrate this concept in greater detail, we 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 Q be a matrix (with a Kronecker-product structure) from this product space. Its tensor representation can be obtained through T = Q.reshape([ m, n, o, m, n, o ]), where the first three axes of the tensor T each correspond to the row axes of matrices from the constituent matrix spaces and the last three axes of the tensor 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]. Remarks Some software libraries use tensors structurally compatible with Kronecker-product matrices to represent quantum states and operations. Operating principles of partial transpose Consider an arbitrary bipartite quantum state defined by its density operator rho = sum(i, j, k, l) R(i, k, j, l) |i><j| |k><l|, where the rank four tensor R(i, k, j, l) determines its coefficients. The order of its indices (axes) follows the tensor representation of matrices with a Kronecker-product structure. Its row indices precede column indices. We define the partial transpose over the second system as pt(rho) = sum(i, j, k, l) R(i, k, j, l) |i><j| |l><k| = sum(i, j, k, l) R(i, l, j, k) |i><j| |k><l|. If the density operator is encoded in a matrix with a Kronecker-product structure, rather than a rank four tensor, 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 Q 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(Q, [ m, n, m, n ]) we obtain a rank four tensor T. Consequently, the partial transpose corresponds to the exchange of the 2nd and the 4th index, 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 and a density matrix rho = sum(i, j, k, l) R(i, k, j, l) |i><j| |k><l|. We define the partial trace performed over its second component as rho' = sum(u) sum(i, j, k, l) R(i, k, j, l) |i><j| <u|k> <l|u> = sum(i, j, k) R(i, k, j, k) |i><j| = sum(i, j) [ sum(k) R(i, k, j, k) ] |i><j|, where we have presumed that the individual { |k>, |l> } kets are orthonormal. If the density operator is encoded in a matrix Q with a Kronecker-product structure, the partial trace can be obtained from its tensor representation T = reshape(Q, [ m, n, m, n ]). 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.