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.

See also

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:

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

For more information about the Wigner transform, see Wigner function on Wikipedia.

Examples

Create a Wigner function.

>>> graph.wigner_transform(np.array([[1]]), [-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]]])