states.isotropic
================

.. py:module:: states.isotropic

.. autoapi-nested-parse::

   Isotropic state is a bipartite quantum state.

   These states are separable for α ≤ 1/(d+1), but are otherwise entangled.



Functions
---------

.. autoapisummary::

   states.isotropic.isotropic


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

.. py:function:: isotropic(dim, alpha)

   Produce a isotropic state :footcite:`Horodecki_1998_Reduction`.

   Returns the isotropic state with parameter :code:`alpha` acting on (:code:`dim`-by-:code:`dim`)-dimensional space.
   The isotropic state has the following form

   .. math::
       \begin{equation}
           \rho_{\alpha} = \frac{1 - \alpha}{d^2} \mathbb{I} \otimes
           \mathbb{I} + \alpha |\psi_+ \rangle \langle \psi_+ | \in
           \mathbb{C}^d \otimes \mathbb{C}^2
       \end{equation}

   where :math:`|\psi_+ \rangle = \frac{1}{\sqrt{d}} \sum_j |j \rangle \otimes |j \rangle` is the maximally entangled
   state.

   .. rubric:: Examples

   To generate the isotropic state with parameter :math:`\alpha=1/2`, we can make the following call to
   :code:`|toqito⟩` as

   .. jupyter-execute::

       from toqito.states import isotropic
       isotropic(3, 1 / 2)

   .. rubric:: References

   .. footbibliography::

   :param dim: The local dimension.
   :param alpha: The parameter of the isotropic state.
   :return: Isotropic state of dimension :code:`dim`.