:py:mod:`state_props.is_mutually_unbiased_basis`
================================================

.. py:module:: state_props.is_mutually_unbiased_basis

.. autoapi-nested-parse::

   Check if states form mutually unbiased basis.



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


Functions
~~~~~~~~~

.. autoapisummary::

   state_props.is_mutually_unbiased_basis.is_mutually_unbiased_basis



.. py:function:: is_mutually_unbiased_basis(vectors)

   Check if list of vectors constitute a mutually unbiased basis :cite:`WikiMUB`.

   We say that two orthonormal bases

   .. math::
       \begin{equation}
           \mathcal{B}_0 = \left\{u_a: a \in \Sigma \right\} \subset \mathbb{C}^{\Sigma}
           \quad \text{and} \quad
           \mathcal{B}_1 = \left\{v_a: a \in \Sigma \right\} \subset \mathbb{C}^{\Sigma}
       \end{equation}

   are *mutually unbiased* if and only if
   :math:`\left|\langle u_a, v_b \rangle\right| = 1/\sqrt{\Sigma}` for all :math:`a, b \in \Sigma`.

   For :math:`n \in \mathbb{N}`, a set of orthonormal bases :math:`\left\{
   \mathcal{B}_0, \ldots, \mathcal{B}_{n-1} \right\}` are mutually unbiased bases if and only if
   every basis is mutually unbiased with every other basis in the set, i.e. :math:`\mathcal{B}_x`
   is mutually unbiased with :math:`\mathcal{B}_x^{\prime}` for all :math:`x \not= x^{\prime}` with
   :math:`x, x^{\prime} \in \Sigma`.

   .. rubric:: Examples

   MUB of dimension :math:`2`.

   For :math:`d=2`, the following constitutes a mutually unbiased basis:

   .. math::
       \begin{equation}
           \begin{aligned}
               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\}. \\
           \end{aligned}
       \end{equation}

   >>> import numpy as np
   >>> from toqito.states import basis
   >>> from toqito.state_props import is_mutually_unbiased_basis
   >>> e_0, e_1 = basis(2, 0), basis(2, 1)
   >>> mub_1 = [e_0, e_1]
   >>> mub_2 = [1 / np.sqrt(2) * (e_0 + e_1), 1 / np.sqrt(2) * (e_0 - e_1)]
   >>> mub_3 = [1 / np.sqrt(2) * (e_0 + 1j * e_1), 1 / np.sqrt(2) * (e_0 - 1j * e_1)]
   >>> nested_mubs = [mub_1, mub_2, mub_3]
   >>> mubs = sum(nested_mubs, [])
   >>> is_mutually_unbiased_basis(mubs)
   True

   Non-MUB of dimension :math:`2`.

   >>> import numpy as np
   >>> from toqito.states import basis
   >>> from toqito.state_props import is_mutually_unbiased_basis
   >>> e_0, e_1 = basis(2, 0), basis(2, 1)
   >>> mub_1 = [e_0, e_1]
   >>> mub_2 = [1 / np.sqrt(2) * (e_0 + e_1), e_1]
   >>> mub_3 = [1 / np.sqrt(2) * (e_0 + 1j * e_1), e_0]
   >>> mubs = [mub_1, mub_2, mub_3]
   >>> is_mutually_unbiased_basis(mubs)
   False

   .. rubric:: References

   .. bibliography::
       :filter: docname in docnames

   :raises ValueError: If at least two vectors are not provided.
   :param vectors: The list of vectors to check.
   :return: :code:`True` if :code:`vec_list` constitutes a mutually unbiased basis, and
            :code:`False` otherwise.