states.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.

Functions

max_mixed(dim[, is_sparse])

Produce the maximally mixed state [1].

Module Contents

states.max_mixed.max_mixed(dim, is_sparse=False)

Produce the maximally mixed state [1].

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

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

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.

Examples

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

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

as follows.

>>> from toqito.states import max_mixed
>>> max_mixed(2, is_sparse=False)
array([[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
>>> max_mixed(2, is_sparse=True) 
<DIAgonal sparse array of dtype 'float64'
    with 2 stored elements (1 diagonals) and shape (2, 2)>

References

[1] (1,2)

Scott Aaronson. Lecture 6: mixed states. URL: https://www.scottaaronson.com/qclec/6.pdf.

Parameters:

• dim (int) – Dimension of the entangled state.

• is_sparse (bool) – True if vector is sparse and False otherwise.

Returns:

The maximally mixed state of dimension dim.

Return type:

[numpy.ndarray, scipy.sparse.dia_array]