rand.random_density_matrix

Generates a random density matrix.

Functions

random_density_matrix(dim[, is_real, k_param, ...])

Generate a random density matrix.

Module Contents

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.

from toqito.rand import random_density_matrix

complex_dm = random_density_matrix(2)

complex_dm

array([[0.71120315-3.98306425e-18j, 0.22304577-3.71349274e-02j],
       [0.22304577+3.71349274e-02j, 0.28879685+3.98306425e-18j]])

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

from toqito.matrix_props import is_density

is_density(complex_dm)

True

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

from toqito.rand import random_density_matrix

real_dm = random_density_matrix(2, is_real=True)

real_dm

array([[0.3962038 , 0.46657396],
       [0.46657396, 0.6037962 ]])

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

from toqito.matrix_props import is_density

is_density(real_dm)

True

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.

from toqito.rand import random_density_matrix

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

bures_mat

array([[ 0.16473398-1.32769492e-18j, -0.12556383-3.31042782e-01j],
       [-0.12556383+3.31042782e-01j,  0.83526602+1.32769492e-18j]])

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

from toqito.matrix_props import is_density

is_density(bures_mat)

True

It is also possible to pass a seed to this function for reproducibility. .. jupyter-execute:

from toqito.rand import random_density_matrix

seeded = random_density_matrix(2, seed=42)

seeded

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

from toqito.matrix_props import is_density

seeded = random_density_matrix(2, seed=42)

is_density(seeded)

True
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) – Default value is equal to dim.

  • distance_metric (str) – The distance metric used to randomly generate the density matrix. This metric is either the Haar measure or the Bures measure. Default value is to use the Haar measure.

  • 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