toqito.matrices.gell_mann¶
Generates the Gell-Mann operator matrices.
Module Contents¶
- toqito.matrices.gell_mann.gell_mann(ind, is_sparse=False)[source]¶
Produce a Gell-Mann operator [@WikiGellMann].
Generates the 3-by-3 Gell-Mann matrix indicated by the value of ind. When ind = 0 gives the identity matrix, while values 1 through 8 each indicate one of the other 8 Gell-Mann matrices.
The 9 Gell-Mann matrices are defined as follows:
- [
- begin{equation}
- begin{aligned}
- lambda_0 = begin{pmatrix}
1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1
end{pmatrix}, quad
- lambda_1 = begin{pmatrix}
0 & 1 & 0 \ 1 & 0 & 0 \ 0 & 0 & 0
end{pmatrix}, quad &
- lambda_2 = begin{pmatrix}
0 & -i & 0 \ i & 0 & 0 \ 0 & 0 & 0
end{pmatrix}, \
- lambda_3 = begin{pmatrix}
1 & 0 & 0 \ 0 & -1 & 0 \ 0 & 0 & 0
end{pmatrix}, quad
- lambda_4 = begin{pmatrix}
0 & 0 & 1 \ 0 & 0 & 0 \ 1 & 0 & 0
end{pmatrix}, quad &
- lambda_5 = begin{pmatrix}
0 & 0 & -i \ 0 & 0 & 0 \ i & 0 & 0
end{pmatrix}, \
- lambda_6 = begin{pmatrix}
0 & 0 & 0 \ 0 & 0 & 1 \ 0 & 1 & 0
end{pmatrix}, quad
- lambda_7 = begin{pmatrix}
0 & 0 & 0 \ 0 & 0 & -i \ 0 & i & 0
end{pmatrix}, quad &
- lambda_8 = frac{1}{sqrt{3}} begin{pmatrix}
1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & -2
end{pmatrix}.
end{aligned}
end{equation}
]
Examples
The Gell-Mann matrix generated from idx = 2 yields the following matrix:
- [
- lambda_2 = begin{pmatrix}
0 & -i & 0 \ i & 0 & 0 \ 0 & 0 & 0
end{pmatrix}
] ```python exec=”1” source=”above” from toqito.matrices import gell_mann
- Raises:
ValueError – Indices must be integers between 0 and 8.
- Parameters:
ind (int) – An integer between 0 and 8 (inclusive).
is_sparse (bool) – Boolean to determine whether array is sparse. Default value is False.
- Return type:
numpy.ndarray | scipy.sparse.csr_array