matrices.gen_gell_mann

Generalized Gell-Mann matrices.

Module Contents

Functions

gen_gell_mann(ind_1, ind_2, dim)

Produce a generalized Gell-Mann operator [1].

matrices.gen_gell_mann.gen_gell_mann(ind_1, ind_2, dim)

Produce a generalized Gell-Mann operator [1].

Construct a dim-by-dim Hermitian operator. These matrices span the entire space of dim-by-dim matrices as ind_1 and ind_2 range from 0 to dim-1, inclusive, and they generalize the Pauli operators when dim = 2 and the Gell-Mann operators when dim = 3.

Examples

The generalized Gell-Mann matrix for ind_1 = 0, ind_2 = 1 and dim = 2 is given as

\[\begin{split}G_{0, 1, 2} = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}.\end{split}\]

This can be obtained in toqito as follows.

>>> from toqito.matrices import gen_gell_mann
>>> gen_gell_mann(0, 1, 2)
array([[0., 1.],
       [1., 0.]])

The generalized Gell-Mann matrix ind_1 = 2, ind_2 = 3, and dim = 4 is given as

\[\begin{split}G_{2, 3, 4} = \begin{pmatrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \end{pmatrix}.\end{split}\]

This can be obtained in toqito as follows.

>>> from toqito.matrices import gen_gell_mann
>>> gen_gell_mann(2, 3, 4)
array([[0., 0., 0., 0.],
       [0., 0., 0., 0.],
       [0., 0., 0., 1.],
       [0., 0., 1., 0.]])

References

[1] (1,2)

Wikipedia. Gell-Mann matrices. https://en.wikipedia.org/wiki/Gell-Mann_matrices.

Parameters:

• ind_1 (int) – A non-negative integer from 0 to dim-1 (inclusive).

• ind_2 (int) – A non-negative integer from 0 to dim-1 (inclusive).

• dim (int) – The dimension of the Gell-Mann operator.

Returns:

The generalized Gell-Mann operator as an array.

Return type:

numpy.ndarray