toqito.state_props.schmidt_rank
===============================

.. py:module:: toqito.state_props.schmidt_rank

.. autoapi-nested-parse::

   Calculate the Schmidt rank of a quantum state.





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

.. py:function:: schmidt_rank(rho, dim = None)

   Compute the Schmidt rank [@WikiScmidtDecomp].

   For complex Euclidean spaces \(\mathcal{X}\) and \(\mathcal{Y}\), a pure state
   \(u \in \mathcal{X} \otimes \mathcal{Y}\) possesses an expansion of the form:

   \[
       u = \sum_{i} \lambda_i v_i w_i,
   \]

   where \(v_i \in \mathcal{X}\) and \(w_i \in \mathcal{Y}\) are orthonormal states.

   The Schmidt coefficients are calculated from

   \[
       A = \text{Tr}_{\mathcal{B}}(u^* u).
   \]

   The Schmidt rank is the number of non-zero eigenvalues of \(A\). The Schmidt rank allows us
   to determine if a given state is entangled or separable. For instance:

   - If the Schmidt rank is 1: The state is separable,
   - If the Schmidt rank > 1: The state is entangled.

   Compute the Schmidt rank of the input `rho`, provided as either a vector or a matrix that
   is assumed to live in bipartite space, where both subsystems have dimension equal to
   `sqrt(len(vec))`.

   The dimension may be specified by the 1-by-2 vector `dim` and the rank in that case is
   determined as the number of Schmidt coefficients larger than `tol`.

   .. rubric:: Examples

   Computing the Schmidt rank of the entangled Bell state should yield a value greater than one.

   ```python exec="1" source="above"
   from toqito.states import bell
   from toqito.state_props import schmidt_rank
   rho = bell(0) @ bell(0).conj().T
   print(schmidt_rank(rho))
   ```


   Computing the Schmidt rank of the entangled singlet state should yield a value greater than
   \(1\).

   ```python exec="1" source="above"
   from toqito.states import bell
   from toqito.state_props import schmidt_rank
   u = bell(2) @ bell(2).conj().T
   print(schmidt_rank(u))
   ```


   Computing the Schmidt rank of a separable state should yield a value equal to \(1\).

   ```python exec="1" source="above"
   from toqito.states import basis
   from toqito.state_props import schmidt_rank
   import numpy as np
   e_0, e_1 = basis(2, 0), basis(2, 1)
   e_00 = np.kron(e_0, e_0)
   e_01 = np.kron(e_0, e_1)
   e_10 = np.kron(e_1, e_0)
   e_11 = np.kron(e_1, e_1)
   rho = 1 / 2 * (e_00 - e_01 - e_10 + e_11)
   rho = rho @ rho.conj().T
   print(schmidt_rank(rho))
   ```

   :param rho: A bipartite vector or matrix to have its Schmidt rank computed.
   :param dim: A 1-by-2 vector or matrix.

   :returns: The Schmidt rank of `rho`.