toqito.states.horodecki
=======================

.. py:module:: toqito.states.horodecki

.. autoapi-nested-parse::

   Horodecki states are bound entangled states.

   These states are entangled, but no pure entangled states can be extracted from these states through local operations and
   classical communication (LOCC).





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

.. py:function:: horodecki(a_param, dim = None)

   Produce a Horodecki state [@Horodecki_1997_Separability][@Chruscinski_2011_OnTheSymmetry].

   Returns the Horodecki state in either \((3 \otimes 3)\)-dimensional space or \((2 \otimes 4)\)-dimensional
   space, depending on the dimensions in the 1-by-2 vector `dim`.

   The Horodecki state was introduced in [1] which serves as an example in \(\mathbb{C}^3 \otimes \mathbb{C}\) or
   \(\mathbb{C}^2 \otimes \mathbb{C}^4\) of an entangled state that is positive under partial transpose (PPT). The
   state is PPT for all \(a \in [0, 1]\) and separable only for `a_param = 0` or `a_param = 1`.

   These states have the following definitions:

   \[
       \begin{equation}
           \rho_a^{3 \otimes 3} = \frac{1}{8a + 1}
           \begin{pmatrix}
               a & 0 & 0 & 0 & a & 0 & 0 & 0 & a \\
               0 & a & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
               0 & 0 & a & 0 & 0 & 0 & 0 & 0 & 0 \\
               0 & 0 & 0 & a & 0 & 0 & 0 & 0 & 0 \\
               0 & 0 & 0 & 0 & a & 0 & 0 & 0 & 0 \\
               0 & 0 & 0 & 0 & 0 & a & 0 & 0 & 0 \\
               0 & 0 & 0 & 0 & 0 & 0 & \frac{1}{2}
               \left( 1 + a \right) & 0 & \frac{1}{2} \sqrt{1 - a^2} \\
               0 & 0 & 0 & 0 & 0 & 0 & 0 & a & 0 \\
               0 & 0 & 0 & 0 & 0 & 0 & \frac{1}{2} \sqrt{1 - a^2} & 0
               & \frac{1}{2} \left(1 + a \right) \\
           \end{pmatrix},
       \end{equation}
   \]

   \[
       \begin{equation}
           \rho_a^{2 \otimes 4} = \frac{1}{7a + 1}
           \begin{pmatrix}
               a & 0 & 0 & 0 & 0 & a & 0 & 0  \\
               0 & a & 0 & 0 & 0 & 0 & a & 0 \\
               0 & 0 & a & 0 & 0 & 0 & 0 & a  \\
               0 & 0 & 0 & a & 0 & 0 & 0 & 0  \\
               0 & 0 & 0 & 0 & \frac{1}{2} \left(1 + a\right) & 0 & 0
               & \frac{1}{2}\sqrt{1 -a^2} \\
               a & 0 & 0 & 0 & 0 & a & 0 & 0 \\
               0 & a & 0 & 0 & 0 & 0 & a & 0 \\
               0 & 0 & a & 0 & \frac{1}{2}\sqrt{1 - a^2} & 0 & 0
               & \frac{1}{2}\left(1 +a \right)
           \end{pmatrix}.
       \end{equation}
   \]

   !!! Note
       Refer to [@Chruscinski_2011_OnTheSymmetry] (specifically equations (1) and (2)) for more information on
       this state and its properties. The 3x3 Horodecki state is defined explicitly in Section 4.1 of
       [@Horodecki_1997_Separability] and the 2x4 Horodecki state is defined explicitly in Section 4.2 of
       [@Horodecki_1997_Separability].

   .. rubric:: Examples

   The following code generates a Horodecki state in \(\mathbb{C}^3 \otimes \mathbb{C}^3\)

   ```python exec="1" source="above"
   from toqito.states import horodecki
   print(horodecki(0.5, [3, 3]))
   ```


   The following code generates a Horodecki state in \(\mathbb{C}^2 \otimes \mathbb{C}^4\).

   ```python exec="1" source="above"
   from toqito.states import horodecki
   print(horodecki(0.5, [2, 4]))
   ```