:py:mod:`state_props.is_pure`
=============================

.. py:module:: state_props.is_pure

.. autoapi-nested-parse::

   Check if state is pure.



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


Functions
~~~~~~~~~

.. autoapisummary::

   state_props.is_pure.is_pure



.. py:function:: is_pure(state)

   Determine if a given state is pure or list of states are pure :cite:`WikiPureSt`.

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

   .. math::
       \rho = u u^*.

   .. rubric:: Examples

   Consider the following Bell state:

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

   The corresponding density matrix of :math:`u` may be calculated by:

   .. math::
       \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 :math:`\rho` yields that the :math:`\rho` is a pure state. This can be
   confirmed in :code:`toqito` as follows:

   >>> from toqito.states import bell
   >>> from toqito.state_props import is_pure
   >>> u = bell(0)
   >>> rho = u * u.conj().T
   >>> is_pure(rho)
   True

   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
   :code:`True` indicating that all states provided to the function are pure.

   >>> 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
   >>>
   >>> is_pure([rho0, rho1, rho2, rho3])
   True

   .. rubric:: References

   .. bibliography::
       :filter: docname in docnames

   :param state: The density matrix representing the quantum state or a list
                 of density matrices representing quantum states.
   :return: :code:`True` if state is pure and :code:`False` otherwise.