toqito.state_props.is_sic_povm
==============================

.. py:module:: toqito.state_props.is_sic_povm

.. autoapi-nested-parse::

   Determine whether a collection of vectors forms a SIC POVM.





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

.. py:function:: is_sic_povm(states, *, tol = 1e-06)

   Check if the provided vectors yield a symmetric informationally complete POVM.

   A set of \(d^2\) unit vectors \(\{\ket{\psi_j}\}\) in \(\mathbb{C}^d\) forms a
   symmetric informationally complete POVM (SIC POVM) when

   \[
       \left| \langle \psi_j, \psi_k \rangle \right|^2 = \frac{1}{d + 1}
       \quad \text{for all } j \neq k,
   \]

   and the projectors satisfy \(\sum_j \ket{\psi_j}\!\bra{\psi_j} = d \mathbb{I}\).

   .. rubric:: Examples

   Qubit tetrahedron SIC.

   ```python exec="1" source="above"
   import numpy as np
   from toqito.state_props import is_sic_povm

   omega = np.exp(2j * np.pi / 3)
   sic_vectors = [
       np.array([0, 1], dtype=np.complex128),
       np.array([np.sqrt(2/3), 1/np.sqrt(3)], dtype=np.complex128),
       np.array([np.sqrt(2/3), omega / np.sqrt(3)], dtype=np.complex128),
       np.array([np.sqrt(2/3), (omega**2) / np.sqrt(3)], dtype=np.complex128),
   ]
   print(is_sic_povm(sic_vectors))
   ```

   Non-SIC vector family.

   ```python exec="1" source="above"
   import numpy as np
   from toqito.state_props import is_sic_povm
   from toqito.states import basis

   e0, e1 = basis(2, 0), basis(2, 1)
   non_sic = [e0, e1, (e0 + e1) / np.sqrt(2), (e0 - e1) / np.sqrt(2)]
   print(is_sic_povm(non_sic))
   ```

   :raises ValueError: If the vectors cannot represent valid quantum states.

   :param states: Collection of vectors to test.
   :param tol: Numerical tolerance used for equality comparisons.

   :returns: `True` when the vectors form a SIC POVM and `False` otherwise.