toqito.states.tile
==================

.. py:module:: toqito.states.tile

.. autoapi-nested-parse::

   Tile state.





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

.. py:function:: tile(idx)

   Produce a Tile state [@Bennett_1999_UPB].

   The Tile states constitute five states on 3-by-3 dimensional space that form a UPB (unextendible product basis).

   Returns one of the following five tile states depending on the value of `idx`:

   \[
       \begin{equation}
           \begin{aligned}
               |\psi_0 \rangle = \frac{1}{\sqrt{2}} |0 \rangle
               \left(|0\rangle - |1\rangle \right),
               \qquad &
               |\psi_1\rangle = \frac{1}{\sqrt{2}}
               \left(|0\rangle - |1\rangle \right) |2\rangle, \\
               |\psi_2\rangle = \frac{1}{\sqrt{2}} |2\rangle
               \left(|1\rangle - |2\rangle \right),
               \qquad &
               |\psi_3\rangle = \frac{1}{\sqrt{2}}
               \left(|1\rangle - |2\rangle \right) |0\rangle, \\
               \qquad &
               |\psi_4\rangle = \frac{1}{3}
               \left(|0\rangle + |1\rangle + |2\rangle)\right)
               \left(|0\rangle + |1\rangle + |2\rangle \right).
           \end{aligned}
       \end{equation}
   \]

   .. rubric:: Examples

   When `idx = 0`, this produces the following tile state

   \[
       \frac{1}{\sqrt{2}} |0\rangle \left( |0\rangle - |1\rangle \right).
   \]

   Using `|toqito⟩`, we can see that this yields the proper state.

   ```python exec="1" source="above"
   from toqito.states import tile
   import numpy as np
   print(tile(0))
   ```

   :raises ValueError: Invalid value for `idx`.

   :param idx: A parameter in [0, 1, 2, 3, 4]

   :returns: Tile state.