# partial_trace¶

Graph.partial_trace(density_matrix, subsystem_dimensions, traced_subsystems, *, name=None)

Calculate the partial trace of a density matrix.

Parameters
• density_matrix (np.ndarray or Tensor) – The density matrix $$\rho$$ of the system to be reduced. Can be a single square matrix or a batch of matrices with dimension (..., D, D).

• subsystem_dimensions (list[int]) – The dimension of each subsystem. The product of the subsystem dimensions is the dimension of the system D.

• traced_subsystems (int or list[int]) – The indices (starting from zero) of the subsystems to be traced out. Each index refers to a different subsystem.

• name (str, optional) – The name of the node.

Returns

The reduced density matrix of shape (..., d, d). The reduced dimension d is equal to the system dimension D divided by the product of the traced out subsystem dimensions.

Return type

Tensor

Notes

Given a density matrix $$\rho$$ of two subsystems $$A$$ and $$B$$, the partial trace over subsystem $$B$$ is defined as

$({\mathrm{Tr}_{B}} \rho)_{ij} = \sum_k \rho_{ik,jk}.$

Examples

>>> graph.partial_trace(np.diag([1, 0, 0, 0]), [2, 2], 1, name="partial")
<Tensor: name="partial", operation_name="partial_trace", shape=(2, 2)>
>>> result = qctrl.functions.calculate_graph(graph=graph, output_node_names=["partial"])
>>> result.output["partial"]["value"]
array([[[1, 0], [0, 0]])

>>> graph.partial_trace(np.eye(10)/10, [2, 5], 1, name="partial")
<Tensor: name="partial", operation_name="partial_trace", shape=(2, 2)>
>>> result = qctrl.functions.calculate_graph(graph=graph, output_node_names=["partial"])
>>> result.output["partial"]["value"]
array([[[0.5, 0], [0, 0.5]])


See more examples in the How to simulate large open system dynamics user guide.