toqito.state_ops.schmidt_decomposition
======================================

.. py:module:: toqito.state_ops.schmidt_decomposition

.. autoapi-nested-parse::

   Schmidt decomposition operation computes the schmidt decomposition of a quantum state or an operator.





Module Contents
---------------

.. py:function:: schmidt_decomposition(rho, dim = None, k_param = 0)

   Compute the Schmidt decomposition of a bipartite vector [@WikiScmidtDecomp].

   .. rubric:: Examples

   Consider the \(3\)-dimensional maximally entangled state:

   \[
       u = \frac{1}{\sqrt{3}} \left( |000 \rangle + |111 \rangle + |222 \rangle \right).
   \]

   We can generate this state using the `|toqito⟩` module as follows.

   ```python exec="1" source="above"
   from toqito.states import max_entangled
   print(max_entangled(3))
   ```

   array([[0.57735027],
       [0.        ],
       [0.        ],
       [0.        ],
       [0.57735027],
       [0.        ],
       [0.        ],
       [0.        ],
       [0.57735027]])

   Computing the Schmidt decomposition of \(u\), we can obtain the corresponding singular values of \(u\) as

   \[
       \frac{1}{\sqrt{3}} \left[1, 1, 1 \right]^{\text{T}}.
   \]

   ```python exec="1" source="above"
   from toqito.states import max_entangled
   from toqito.state_ops import schmidt_decomposition

   singular_vals, u_mat, vt_mat = schmidt_decomposition(max_entangled(3))

   matrices = {
       "Singular values": singular_vals,
       "U matrix": u_mat,
       "V^T matrix": vt_mat,
   }

   for name, mat in matrices.items():
       print(f"{name}:\n{mat}\n")
   ```

   :raises ValueError: If matrices are not of equal dimension.

   :param rho: A bipartite quantum state to compute the Schmidt decomposition of.
   :param dim: An array consisting of the dimensions of the subsystems (default gives subsystems
               equal dimensions).
   :param k_param: How many terms of the Schmidt decomposition should be computed (default is 0).

   :returns: The Schmidt decomposition of the `rho` input.