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gen_pauli_z

Produces a generalized Pauli-Z operator matrix.

gen_pauli_z 

gen_pauli_z(dim: int) -> ndarray

Produce gen_pauli_z matrix 1.

Returns the gen_pauli_z matrix of dimension dim described in 1. The gen_pauli_z matrix generates the following dim-by-dim matrix

\[ \Sigma_{1, d} = \begin{pmatrix} 1 & 0 & 0 & \ldots & 0 \\ 0 & \omega & 0 & \ldots & 0 \\ 0 & 0 & \omega^2 & \ldots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \ldots & \omega^{d-1} \end{pmatrix} \]

where \(\omega\) is the n-th primitive root of unity.

The gen_pauli_z matrix is primarily used in the construction of the generalized Pauli operators.

Parameters:

  • dim (int) –

    Dimension of the matrix.

Returns:

  • ndarray –

    dim-by-dim gen_pauli_z matrix.

Examples:

The gen_pauli_z matrix generated from \(d = 3\) yields the following matrix:

\[ \Sigma_{1, 3} = \begin{pmatrix} 1 & 0 & 0 \\ 0 & \omega & 0 \\ 0 & 0 & \omega^2 \end{pmatrix} \]
from toqito.matrices import gen_pauli_z

print(gen_pauli_z(3))

[[ 1. +0.j         0. +0.j         0. +0.j       ]
 [ 0. +0.j        -0.5+0.8660254j  0. +0.j       ]
 [ 0. +0.j         0. +0.j        -0.5-0.8660254j]]

References

1 Wikipedia. Generalizations of Pauli matrices. link.

Source code in toqito/matrices/gen_pauli_z.py 
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def gen_pauli_z(dim: int) -> np.ndarray:
    r"""Produce gen_pauli_z matrix [@wikipediaclock].

    Returns the gen_pauli_z matrix of dimension `dim` described in [@wikipediaclock].
    The gen_pauli_z matrix generates the following `dim`-by-`dim` matrix

    \[
        \Sigma_{1, d} = \begin{pmatrix}
                        1 & 0 & 0 & \ldots & 0 \\
                        0 & \omega & 0 & \ldots & 0 \\
                        0 & 0 & \omega^2 & \ldots & 0 \\
                        \vdots & \vdots & \vdots & \ddots & \vdots \\
                        0 & 0 & 0 & \ldots & \omega^{d-1}
                   \end{pmatrix}
    \]

    where \(\omega\) is the n-th primitive root of unity.

    The gen_pauli_z matrix is primarily used in the construction of the generalized
    Pauli operators.

    Args:
        dim: Dimension of the matrix.

    Returns:
        `dim`-by-`dim` gen_pauli_z matrix.

    Examples:
        The gen_pauli_z matrix generated from \(d = 3\) yields the following matrix:

        \[
            \Sigma_{1, 3} = \begin{pmatrix}
                1 & 0 & 0 \\
                0 & \omega & 0 \\
                0 & 0 & \omega^2
            \end{pmatrix}
        \]

        ```python exec="1" source="above" result="text"
        from toqito.matrices import gen_pauli_z

        print(gen_pauli_z(3))
        ```

    """
    c_var = 2j * pi / dim
    omega = (exp(k * c_var) for k in range(dim))
    return np.diag(list(omega))