toqito.state_props.is_pure
==========================

.. py:module:: toqito.state_props.is_pure

.. autoapi-nested-parse::

   Checks if a quantum state is a pure state.





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

.. py:function:: is_pure(state)

   Determine if a given state is pure or list of states are pure [@WikiPureSt].

   A state is said to be pure if it is a density matrix with rank equal to 1. Equivalently, the
   state \(\rho\) is pure if there exists a unit vector \(u\) such that:

   \[
       \rho = u u^*.
   \]

   .. rubric:: Examples

   Consider the following Bell state:

   \[
       u = \frac{1}{\sqrt{2}} \left( |00 \rangle + |11 \rangle \right) \in \mathcal{X}.
   \]

   The corresponding density matrix of \(u\) may be calculated by:

   \[
       \rho = u u^* = \frac{1}{2} \begin{pmatrix}
                        1 & 0 & 0 & 1 \\
                        0 & 0 & 0 & 0 \\
                        0 & 0 & 0 & 0 \\
                        1 & 0 & 0 & 1
                      \end{pmatrix} \in \text{D}(\mathcal{X}).
   \]

   Calculating the rank of \(\rho\) yields that the \(\rho\) is a pure state. This can be
   confirmed in `|toqito⟩` as follows:

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

   It is also possible to determine whether a set of density matrices are pure. For instance, we
   can see that the density matrices corresponding to the four Bell states yield a result of
   `True` indicating that all states provided to the function are pure.

   ```python exec="1" source="above"
   from toqito.states import bell
   from toqito.state_props import is_pure
   u0, u1, u2, u3 = bell(0), bell(1), bell(2), bell(3)
   rho0 = u0 @ u0.conj().T
   rho1 = u1 @ u1.conj().T
   rho2 = u2 @ u2.conj().T
   rho3 = u3 @ u3.conj().T
   print(is_pure([rho0, rho1, rho2, rho3]))
   ```

   :param state: The density matrix representing the quantum state or a list of density matrices representing quantum
   :param states.:

   :returns: `True` if state is pure and `False` otherwise.