# wigner_transform

Graph.wigner_transform(density_matrix, position, momentum, offset=0, *, name=None)

Transform a density matrix into a Wigner function (or a batch of them).

Parameters:
• density_matrix (Tensor or np.ndarray) – The density matrix $$\rho$$ in the Fock basis. Must be of shape (..., D, D).

• position (list or tuple or np.ndarray) – The dimensionless position vector $$q$$. Must be a 1D array of shape (L,).

• momentum (list or tuple or np.ndarray) – The dimensionless momentum vector $$p$$. Must be a 1D array of shape (K,).

• offset (int, optional) – The lowest Fock state. Defaults to 0.

• name (str or None, optional) – The name of the node.

Returns:

The Wigner function with shape (..., L, K).

Return type:

Tensor

Warning

This function currently does not support calculating the gradient with respect to its inputs. Therefore, it cannot be used in a graph for a calculate_optimization or calculate_stochastic_optimization call, which will raise a RuntimeError. Please use gradient-free optimization if you want to perform an optimization task with this function. You can learn more about it in the How to optimize controls using gradient-free optimization user guide.

coherent_state

Create a coherent state (or a batch of them).

fock_state

Create a Fock state (or a batch of them).

Notes

The Wigner transform is defined as:

$W(q,p) = \frac{1}{2\pi} \int_{-\infty}^{\infty} \mathrm{e}^{\mathrm{i}sp} \langle q - s/2| \rho | q + s/2 \rangle \mathrm{d}s .$

Examples

Create a Wigner function.

>>> graph.wigner_transform(np.array([]), [-1, 0, 1], np.zeros(1), name="wigner")
<Tensor: name="wigner", operation_name="wigner_transform", shape=(3, 1)>
>>> result = qctrl.functions.calculate_graph(graph=graph, output_node_names=["wigner"])
>>> result.output["wigner"]["value"]
array([[ 0.11709966+0.j],
[ 0.31830989+0.j],
[ 0.11709966+0.j]])


Create a batch of Wigner function.

>>> graph.wigner_transform(
np.array([[[1, 0], [0, 0]], [[0, 0], [0, 1]]]),
np.array([-1, 0, 1]),
np.zeros(1),
name="wigner_batch",
)
<Tensor: name="wigner_batch", operation_name="wigner_transform", shape=(2, 3, 1)>
>>> result = qctrl.functions.calculate_graph(
...     graph=graph, output_node_names=["wigner_batch"]
... )
>>> result.output["wigner_batch"]["value"]
array([[[ 0.11709966+0.j],
[ 0.31830989+0.j],
[ 0.11709966+0.j]],
[[ 0.11709966+0.j],
[-0.31830989+0.j],
[ 0.11709966+0.j]]])