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max_mixed

Maximally mixed states are states which are formed as a uniform mixture of states in an orthonormal basis.

The density matrix of a maximally mixed state is directly proportional to the identity matrix.

max_mixed 

max_mixed(
    dim: int, is_sparse: bool = False
) -> ndarray | dia_array

Produce the maximally mixed state 1.

Produces the maximally mixed state on of dim dimensions. The maximally mixed state is defined as

\[ \omega = \frac{1}{d} \begin{pmatrix} 1 & 0 & \ldots & 0 \\ 0 & 1 & \ldots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \ldots & 1 \end{pmatrix}, \]

or equivalently, it is defined as

\[ \omega = \frac{\mathbb{I}}{\text{dim}(\mathcal{X})} \]

for some complex Euclidean space \(\mathcal{X}\). The maximally mixed state is sometimes also referred to as the tracial state.

The maximally mixed state is returned as a sparse matrix if is_sparse = True and is full if is_sparse = False.

Parameters:

  • dim (int) –

    Dimension of the entangled state.

  • is_sparse (bool, default: False ) –

    True if vector is sparse and False otherwise.

Returns:

  • ndarray | dia_array –

    The maximally mixed state of dimension dim.

Examples:

Using |toqito⟩, we can generate the \(2\)-dimensional maximally mixed state

\[ \omega_2 = \frac{1}{2} \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \]

as follows.

from toqito.states import max_mixed
print(max_mixed(2, is_sparse=False))

[[0.5 0. ]
 [0.  0.5]]

One may also generate a maximally mixed state returned as a sparse matrix

from toqito.states import max_mixed
print(max_mixed(2, is_sparse=True))

<DIAgonal sparse array of dtype 'float64'
    with 2 stored elements (1 diagonals) and shape (2, 2)>
  Coords    Values
  (0, 0)    0.5
  (1, 1)    0.5

References

1 Aaronson, Scott. Lecture 6: Mixed States. link.

Source code in toqito/states/max_mixed.py 
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def max_mixed(dim: int, is_sparse: bool = False) -> np.ndarray | dia_array:
    r"""Produce the maximally mixed state [@aaronson2018mixed].

    Produces the maximally mixed state on of `dim` dimensions. The maximally mixed state is defined as

    \[
        \omega = \frac{1}{d} \begin{pmatrix}
                        1 & 0 & \ldots & 0 \\
                        0 & 1 & \ldots & 0 \\
                        \vdots & \vdots & \ddots & \vdots \\
                        0 & 0 & \ldots & 1
                    \end{pmatrix},
    \]

    or equivalently, it is defined as

    \[
        \omega = \frac{\mathbb{I}}{\text{dim}(\mathcal{X})}
    \]

    for some complex Euclidean space \(\mathcal{X}\). The maximally mixed state is sometimes also referred to as the
    tracial state.

    The maximally mixed state is returned as a sparse matrix if `is_sparse = True` and is full if `is_sparse
    = False`.

    Args:
        dim: Dimension of the entangled state.
        is_sparse: `True` if vector is sparse and `False` otherwise.

    Returns:
        The maximally mixed state of dimension `dim`.

    Examples:
        Using `|toqito⟩`, we can generate the \(2\)-dimensional maximally mixed state

        \[
            \omega_2 = \frac{1}{2}
            \begin{pmatrix}
                1 & 0 \\
                0 & 1
            \end{pmatrix}
        \]

        as follows.

        ```python exec="1" source="above" result="text"
        from toqito.states import max_mixed
        print(max_mixed(2, is_sparse=False))
        ```



        One may also generate a maximally mixed state returned as a sparse matrix

        ```python exec="1" source="above" result="text"
        from toqito.states import max_mixed
        print(max_mixed(2, is_sparse=True))
        ```

    """
    if is_sparse:
        return 1 / dim * eye_array(dim)
    return 1 / dim * np.eye(dim)