toqito.states.bell
==================

.. py:module:: toqito.states.bell

.. autoapi-nested-parse::

   Bell states represent the simplest examples of quantum entanglement of two qubits.

   Also known as EPR pairs, Bell states comprise of four quantum states in a superposition of 0 and 1.





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

.. py:function:: bell(idx)

   Produce a Bell state [@WikiBellSt].

   Returns one of the following four Bell states depending on the value of `idx`:

   \[
       \begin{equation}
           \begin{aligned}
               u_0 = \frac{1}{\sqrt{2}} \left( |00 \rangle + |11 \rangle \right), &
               \qquad &
               u_1 = \frac{1}{\sqrt{2}} \left( |00 \rangle - |11 \rangle \right), \\
               u_2 = \frac{1}{\sqrt{2}} \left( |01 \rangle + |10 \rangle \right), &
               \qquad &
               u_3 = \frac{1}{\sqrt{2}} \left( |01 \rangle - |10 \rangle \right).
           \end{aligned}
       \end{equation}
   \]

   .. rubric:: Examples

   When `idx = 0`, this produces the following Bell state:

   \[
       u_0 = \frac{1}{\sqrt{2}} \left( |00 \rangle + |11 \rangle \right).
   \]

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

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

   :raises ValueError: If `idx` is not an integer.

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

   :returns: Bell state with index `idx`.