def random_density_matrix(
dim: int,
is_real: bool = False,
k_param: Union[List[int], int] = None,
distance_metric: str = "haar",
) -> np.ndarray:
"""
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.
:param dim: The number of rows (and columns) of the density matrix.
:param is_real: Boolean denoting whether the returned matrix will have all
real entries or not.
:param k_param: Default value is equal to `dim`.
:param distance_metric: 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.
:return: A `dim`-by-`dim` random density matrix.
"""
if k_param is None:
k_param = dim
# Haar / Hilbert-Schmidt measure.
gin = np.random.rand(dim, k_param)
if not is_real:
gin = gin + 1j * np.random.rand(dim, k_param)
if distance_metric == "bures":
gin = np.matmul(random_unitary(dim, is_real) + np.identity(dim), gin)
rho = np.matmul(gin, np.matrix(gin).H)
return np.divide(rho, np.trace(rho))
def test_is_unitary(self):
"""Test that unitary matrix returns True."""
mat = random_unitary(2)
self.assertEqual(is_unitary(mat), True)
def test_random_unitary_not_real(self):
"""Generate random non-real unitary matrix."""
mat = random_unitary(2)
self.assertEqual(is_unitary(mat), True)
def test_random_unitary_vec_dim(self):
"""Generate random non-real unitary matrix."""
mat = random_unitary([4, 4], True)
self.assertEqual(is_unitary(mat), True)
def random_density_matrix(
dim: int,
is_real: bool = False,
k_param: Union[List[int], int] = None,
distance_metric: str = "haar",
) -> np.ndarray:
r"""
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 :math:`n`-
dimensional density matrix. For :math:`d=2`, this can be accomplished as
follows.
>>> from toqito.random.random_density_matrix import random_density_matrix
>>> complex_dm = random_density_matrix(2)
>>> complex_dm
[[0.34903796+0.j 0.4324904 +0.103298j]
[0.4324904 -0.103298j 0.65096204+0.j ]]
We can verify that this is in fact a valid density matrix using the
`is_denisty` function from `toqito` as follows
>>> from toqito.linear_algebra.properties.is_density import is_density
>>> is_density(complex_dm)
True
We can also generate random density matrices that are real-valued as
follows.
>>> from toqito.random.random_density_matrix import random_density_matrix
>>> real_dm = random_density_matrix(2, is_real=True)
>>> real_dm
[[0.37330805 0.46466224]
[0.46466224 0.62669195]]
Again, verifying that this is a valid density matrix can be done as follows.
>>> from toqito.linear_algebra.properties.is_density 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.random.random_density_matrix import random_density_matrix
>>> bures_mat = random_density_matrix(2, distance_metric="bures")
>>> bures_mat
[[0.59937164+0.j 0.45355087-0.18473365j]
[0.45355087+0.18473365j 0.40062836+0.j ]]
As before, we can verify that this matrix generated is a valid density
matrix.
>>> from toqito.linear_algebra.properties.is_density import is_density
>>> is_density(bures_mat)
True
:param dim: The number of rows (and columns) of the density matrix.
:param is_real: Boolean denoting whether the returned matrix will have all
real entries or not.
:param k_param: Default value is equal to `dim`.
:param distance_metric: 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.
:return: A `dim`-by-`dim` random density matrix.
"""
if k_param is None:
k_param = dim
# Haar / Hilbert-Schmidt measure.
gin = np.random.rand(dim, k_param)
if not is_real:
gin = gin + 1j * np.random.rand(dim, k_param)
if distance_metric == "bures":
gin = np.matmul(random_unitary(dim, is_real) + np.identity(dim), gin)
rho = np.matmul(gin, np.matrix(gin).H)
return np.divide(rho, np.trace(rho))