toqito.state_props.is_mutually_orthogonal
=========================================

.. py:module:: toqito.state_props.is_mutually_orthogonal

.. autoapi-nested-parse::

   Checks if quantum states are mutually orthogonal.





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

.. py:function:: is_mutually_orthogonal(vec_list)

   Check if list of vectors are mutually orthogonal [@WikiOrthog].

   We say that two bases

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

   are *mutually orthogonal* if and only if
   \(\left|\langle u_a, v_b \rangle\right| = 0\) for all \(a, b \in \Sigma\).

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

   .. rubric:: Examples

   The Bell states constitute a set of mutually orthogonal vectors.

   ```python exec="1" source="above"
   from toqito.states import bell
   from toqito.state_props import is_mutually_orthogonal
   states = [bell(0), bell(1), bell(2), bell(3)]
   print(is_mutually_orthogonal(states))
   ```


   The following is an example of a list of vectors that are not mutually orthogonal.

   ```python exec="1" source="above"
   import numpy as np
   from toqito.states import bell
   from toqito.state_props import is_mutually_orthogonal
   states = [np.array([1, 0]), np.array([1, 1])]
   print(is_mutually_orthogonal(states))
   ```

   :raises ValueError: If at least two vectors are not provided.

   :param vec_list: The list of vectors to check.

   :returns: `True` if `vec_list` are mutually orthogonal, and `False` otherwise.