toqito.states.mutually_unbiased_basis
=====================================

.. py:module:: toqito.states.mutually_unbiased_basis

.. autoapi-nested-parse::

   Mutually unbiased basis states.

   If a system prepared in an eigenstate of one of the bases gives an equal probability of (1/d) when measured with respect
   to the other bases, mutually unbiased basis states are orthonormal bases in the Hilbert space Cᵈ.





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

.. py:function:: mutually_unbiased_basis(dim)

   Generate list of MUBs for a given dimension [@WikiMUB].

   Note that this function only works if the dimension provided is prime or a power of a prime. Otherwise, we don't
   know how to generate general MUBs.

   .. rubric:: Examples

   For the case of dimension 2, the three mutually unbiased bases are provided by:

   \[
       M_0 = \left\{|0\rangle, |1\rangle \right\}, \\
       M_1 = \left\{\frac{|0\rangle + |1\rangle}{\sqrt{2}}, \frac{|0\rangle - |1\rangle}{\sqrt{2}}\right\}
       M_2 = \left\{\frac{|0\rangle + i|1\rangle}{\sqrt{2}}, \frac{|0\rangle - i|1\rangle}{\sqrt{2}}\right\}
   \]

   The six vectors above are obtained accordingly:

   ```python exec="1" source="above" session="mubs"
   from toqito.states import mutually_unbiased_basis
   mubs = mutually_unbiased_basis(2)
   print(len(mubs))
   ```

   ```python exec="1" source="above" session="mubs"
   lst =[vec.shape for vec in mubs]
   print(lst)
   ```

   :param dim: The dimension of the mutually unbiased bases to produce.

   :returns: The set of mutually unbiased bases of dimension `dim` (if known).