toqito.matrix_props.is_density

Checks if the matrix is a density matrix.

Module Contents

toqito.matrix_props.is_density.is_density(mat)

Check if matrix is a density matrix [@WikiDen].

A matrix is a density matrix if its trace is equal to one and it has the property of being positive semidefinite (PSD).

Examples

Consider the Bell state:

[

u = frac{1}{sqrt{2}} |00 rangle + frac{1}{sqrt{2}} |11 rangle.

]

Constructing the matrix (rho = u u^*) defined as

[

rho = frac{1}{2} begin{pmatrix}

1 & 0 & 0 & 1 \ 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 \ 1 & 0 & 0 & 1

end{pmatrix}

]

our function indicates that this is indeed a density operator as the trace of (rho) is equal to (1) and the matrix is positive semidefinite.

`python exec="1" source="above" from toqito.matrix_props import is_density from toqito.states import bell import numpy as np rho = bell(0) @ bell(0).conj().T print(is_density(rho)) `

Alternatively, the following example matrix (sigma) defined as

[

sigma = frac{1}{2} begin{pmatrix}

1 & 2 \ 3 & 1

end{pmatrix}

]

does satisfy (text{Tr}(sigma) = 1), however fails to be positive semidefinite, and is therefore not a density operator. This can be illustrated using |toqito⟩ as follows.

```python exec=”1” source=”above” import numpy as np from toqito.states import bell from toqito.matrix_props import is_density

sigma = 1/2 * np.array([[1, 2], [3, 1]])

print(is_density(sigma)) ```

Parameters:

mat (numpy.ndarray) – Matrix to check.

Returns:

Return True if matrix is a density matrix, and False otherwise.

Return type:

bool