toqito.matrix_props.positive_semidefinite_rank¶

Calculates the positive semidefinite rank of a nonnegative matrix.

Module Contents¶

toqito.matrix_props.positive_semidefinite_rank.positive_semidefinite_rank(mat, max_rank=10)[source]¶

Compute the positive semidefinite rank (PSD rank) of a nonnegative matrix.

The definition of PSD rank is defined in [@Fawzi_2015_Positive].

Finds the PSD rank of an input matrix by checking feasibility for increasing rank.

Examples

As an example (Equation 21 from [@Heinosaari_2024_Can]), the PSD rank of the following matrix

[

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

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

end{pmatrix}

]

is known to be (text{rank}_{text{PSD}}(A) = 2).

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

print(positive_semidefinite_rank(1/2 * np.array([[0, 1, 1], [1,0,1], [1,1,0]]))) ```

The PSD rank of the identity matrix is the dimension of the matrix [@Fawzi_2015_Positive].

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

print(positive_semidefinite_rank(np.identity(3))) ```

Parameters:

Return type:

int | None