qnet.algebra.core.matrix_algebra module

Matrices of Operators

Summary

Classes:

Matrix

Matrix of Expressions

Functions:

block_matrix	Generate the operator matrix with quadrants
diagm	Generalizes the diagonal matrix creation capabilities of numpy.diag to Matrix objects.
hstackm	Generalizes numpy.hstack to Matrix objects.
identity_matrix	Generate the N-dimensional identity matrix.
permutation_matrix	Return orthogonal permutation matrix for permutation tuple
vstackm	Generalizes numpy.vstack to Matrix objects.
zerosm	Generalizes numpy.zeros to Matrix objects.

__all__: Matrix, block_matrix, diagm, hstackm, identity_matrix, vstackm, zerosm

Reference

class qnet.algebra.core.matrix_algebra.Matrix(m)

Bases: qnet.algebra.core.abstract_algebra.Expression

Matrix of Expressions

Matrices of Operator expressions are required for the SLH formalism.

matrix = None

shape

The shape of the matrix (nrows, ncols)

block_structure

For square matrices this gives the block (-diagonal) structure of the matrix as a tuple of integers that sum up to the full dimension.

Return type:tuple

args

The tuple of positional arguments for the instantiation of the Expression

is_zero

Are all elements of the matrix zero?

transpose()

The transpose matrix

conjugate()

The element-wise conjugate matrix

This is defined only if all the entries in the matrix have a defined conjugate (i.e., they have a conjugate method). This is not the case for a matrix of operators. In such a case, only an elementwise() adjoint() would be applicable, but this is mathematically different from a complex conjugate.

Raises:NoConjugateMatrix – if any entries have no conjugate method

real

Element-wise real part

Raises:NoConjugateMatrix – if entries have no conjugate method and no other way to determine the real part

Note

A mathematically equivalent way to obtain a real matrix from a complex matrix M is:

(M.conjugate() + M) / 2

However, the result may not be identical to M.real, as the latter tries to convert elements of the matrix to real values directly, if possible, and only uses the conjugate as a fall-back

imag

Element-wise imaginary part

Raises:NoConjugateMatrix – if entries have no conjugate method and no other way to determine the imaginary part

Note

A mathematically equivalent way to obtain an imaginary matrix from a complex matrix M is:

(M.conjugate() - M) / (I * 2)

with same same caveats as real.

T

Alias for transpose()

adjoint()

Adjoint of the matrix

This is the transpose and the Hermitian adjoint of all elements.

dag()

Adjoint of the matrix

This is the transpose and the Hermitian adjoint of all elements.

trace()

H

Alias for adjoint()

element_wise(func, *args, **kwargs)

Apply a function to each matrix element and return the result in a new operator matrix of the same shape.

Parameters:

• func (FunctionType) – A function to be applied to each element. It must take the element as its first argument.
• args – Additional positional arguments to be passed to func
• kwargs – Additional keyword arguments to be passed to func

Returns:

Matrix with results of func, applied element-wise.

Return type:

Matrix

series_expand(param, about, order)

Expand the matrix expression as a truncated power series in a scalar parameter.

Parameters:

• param (Symbol) – Expansion parameter.
• about (Scalar) – Point about which to expand.
• order (int) – Maximum order of expansion >= 0

Returns:

tuple of length (order+1), where the entries are the expansion coefficients.

expand()

Expand each matrix element distributively.

Returns:Expanded matrix. Return type:Matrix

free_symbols

Set of free SymPy symbols contained within the expression.

space

Combined Hilbert space of all matrix elements.

simplify_scalar(func=<function simplify>)

Simplify all scalar expressions appearing in the Matrix.

qnet.algebra.core.matrix_algebra.hstackm(matrices)

Generalizes numpy.hstack to Matrix objects.

qnet.algebra.core.matrix_algebra.vstackm(matrices)

Generalizes numpy.vstack to Matrix objects.

qnet.algebra.core.matrix_algebra.diagm(v, k=0)

Generalizes the diagonal matrix creation capabilities of numpy.diag to Matrix objects.

qnet.algebra.core.matrix_algebra.block_matrix(A, B, C, D)

Generate the operator matrix with quadrants

\[\begin{split}\begin{pmatrix} A B \\ C D \end{pmatrix}\end{split}\]

Parameters:

• A (Matrix) – Matrix of shape (n, m)
• B (Matrix) – Matrix of shape (n, k)
• C (Matrix) – Matrix of shape (l, m)
• D (Matrix) – Matrix of shape (l, k)

Returns:

The combined block matrix [[A, B], [C, D]].

Return type:

Matrix

qnet.algebra.core.matrix_algebra.identity_matrix(N)

Generate the N-dimensional identity matrix.

Parameters:N (int) – Dimension Returns:Identity matrix in N dimensions Return type:Matrix

qnet.algebra.core.matrix_algebra.zerosm(shape, *args, **kwargs)

Generalizes numpy.zeros to Matrix objects.

qnet.algebra.core.matrix_algebra.permutation_matrix(permutation)

Return orthogonal permutation matrix for permutation tuple

Return an orthogonal permutation matrix \(M_\sigma\) for a permutation \(\sigma\) defined by the image tuple \((\sigma(1), \sigma(2),\dots \sigma(n))\), such that

\[M_\sigma \vec{e}_i = \vec{e}_{\sigma(i)}\]

where \(\vec{e}_k\) is the k-th standard basis vector. This definition ensures a composition law:

\[M_{\sigma \cdot \tau} = M_\sigma M_\tau.\]

The column form of \(M_\sigma\) is thus given by

\[M = ( \vec{e}_{\sigma(1)}, \vec{e}_{\sigma(2)}, \dots \vec{e}_{\sigma(n)}).\]

Parameters:permutation (tuple) – A permutation image tuple (zero-based indices!)