density_matrix_expectation_value

Graph.density_matrix_expectation_value(density_matrix, operator, *, name=None)

Calculate the expectation value of an operator with respect to a density matrix.

The last two dimensions of the density matrix must be equal to the last two dimensions of the operator and their batch shapes must be broadcastable.

Parameters:

density_matrix (np.ndarray or Tensor) – The density matrix. It must be of shape (..., D, D).
operator (np.ndarray or Tensor) – The operator. It must be of shape (..., D, D).
name (str or None, optional) – The name of the node.

Returns:

The expectation value with shape (...).

Return type:

Tensor

See also

expectation_value: Expectation value of an operator with respect to a pure state.
inner_product: Inner product of two vectors.
outer_product: Outer product of two vectors.
trace: Trace of an object.

Notes

The expectation value of an operator \(\mathbf{A}\) with respect to a density matrix \(\rho=\sum_i p_i |\psi_i\rangle\langle\psi_i|\) is defined as

\[{\mathrm{Tr}}(A\rho) = {\mathrm{Tr}}(A\sum_i p_i |\psi_i\rangle\langle\psi_i|) = \sum_i p_i \langle\psi_i|A|\psi_i\rangle .\]

For more information about the density matrix expectation value, see density matrix on Wikipedia.

Examples

>>> graph.density_matrix_expectation_value(
...     np.array([[0.9, 0.], [0., 0.1]]), np.array([[1., 0.], [0., -1.]]), name="expectation"
... )
<Tensor: name="expectation", operation_name="density_matrix_expectation_value", shape=()>
>>> result = qctrl.functions.calculate_graph(graph=graph, output_node_names=["expectation"])
>>> result.output["expectation"]["value"]
0.8
>>> graph.density_matrix_expectation_value(
...     np.array([[0.9, 0.], [0., 0.1]]),
...     np.array([[[0., 1.], [1., 0.]], [[0., -1.j], [1.j, 0.]], [[1., 0.], [0., -1.]]]),
...     name="expectation"
... )
<Tensor: name="expectation", operation_name="expectation_value", shape=(3,)>
>>> result = qctrl.functions.calculate_graph(graph=graph, output_node_names=["expectation"])
>>> result.output["expectation"]["value"]
array([0. +0.j, 0. +0.j, 0.8+0.j])