toqito.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.

Module Contents

toqito.states.max_mixed.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.

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)) `

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