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

.. py:module:: toqito.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.





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

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

   Produce a isotropic state [@Horodecki_1998_Reduction].

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

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

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

   .. rubric:: Examples

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

   ```python exec="1" source="above"
   from toqito.states import isotropic
   print(isotropic(3, 1 / 2))
   ```

   :param dim: The local dimension.
   :param alpha: The parameter of the isotropic state.

   :returns: Isotropic state of dimension `dim`.