toqito.rand.random_density_matrix

Generates a random density matrix.

Module Contents

toqito.rand.random_density_matrix.random_density_matrix(dim, is_real=False, k_param=None, distance_metric='haar', seed=None)

Generate a random density matrix.

Generates a random dim-by-dim density matrix distributed according to the Hilbert-Schmidt measure. The matrix is of rank <= k_param distributed according to the distribution distance_metric If is_real = True, then all of its entries will be real. The variable distance_metric must be one of:

• haar (default):

  Generate a larger pure state according to the Haar measure and trace out the extra dimensions. Sometimes called the Hilbert-Schmidt measure when k_param = dim.

• bures:

  The Bures measure.

Examples

Using |toqito⟩, we may generate a random complex-valued (n)- dimensional density matrix. For (d=2), this can be accomplished as follows.

```python exec=”1” source=”above” session=”complex_dm_example” from toqito.rand import random_density_matrix

complex_dm = random_density_matrix(2)

print(complex_dm) ```

We can verify that this is in fact a valid density matrix using the is_density function from |toqito⟩ as follows

```python exec=”1” source=”above” session=”complex_dm_example” from toqito.matrix_props import is_density

print(is_density(complex_dm)) ```

We can also generate random density matrices that are real-valued as follows.

```python exec=”1” source=”above” session=”real_dm_example” from toqito.rand import random_density_matrix

real_dm = random_density_matrix(2, is_real=True)

print(real_dm) ```

Again, verifying that this is a valid density matrix can be done as follows.

```python exec=”1” source=”above” session=”real_dm_example” from toqito.matrix_props import is_density

print(is_density(real_dm)) ```

By default, the random density operators are constructed using the Haar measure. We can select to generate the random density matrix according to the Bures metric instead as follows.

```python exec=”1” source=”above” session=”bures_dm_example” from toqito.rand import random_density_matrix

bures_mat = random_density_matrix(2, distance_metric=”bures”)

print(bures_mat) ```

As before, we can verify that this matrix generated is a valid density matrix.

```python exec=”1” source=”above” session=”bures_dm_example” from toqito.matrix_props import is_density

print(is_density(bures_mat)) ```

It is also possible to pass a seed to this function for reproducibility. ```python exec=”1” source=”above” session=”seeded_dm_example” from toqito.rand import random_density_matrix

seeded = random_density_matrix(2, seed=42)

print(seeded) ```

We can once again verify that this is in fact a valid density matrix using the is_density function from |toqito⟩ as follows

```python exec=”1” source=”above” session=”seeded_dm_example” from toqito.matrix_props import is_density

seeded = random_density_matrix(2, seed=42)

print(is_density(seeded)) ```

Parameters:

• dim (int) – The number of rows (and columns) of the density matrix.

• is_real (bool) – Boolean denoting whether the returned matrix will have all real entries or not.

• k_param (list[int] | int | None) – Default value is equal to dim.

• distance_metric (str) – The distance metric used to randomly generate the density matrix. This metric is either the

• measure. (Haar measure or the Bures measure. Default value is to use the Haar)

• seed (int | None) – A seed used to instantiate numpy’s random number generator.

Returns:

A dim-by-dim random density matrix.

Return type:

numpy.ndarray