toqito.states.max_mixed
=======================

.. py:module:: toqito.states.max_mixed

.. autoapi-nested-parse::

   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.





Module Contents
---------------

.. py:function:: max_mixed(dim, is_sparse = False)

   Produce the maximally mixed state [@Aaronson_2018_MaxMixed].

   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`.

   .. rubric:: 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"
   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"
   from toqito.states import max_mixed
   print(max_mixed(2, is_sparse=True))
   ```

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

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