toqito.matrix_props.is_ldoi
===========================

.. py:module:: toqito.matrix_props.is_ldoi

.. autoapi-nested-parse::

   Check if a quantum state is LDOI (Local Diagonal Orthogonal Invariant).





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

.. py:function:: is_ldoi(mat, rtol = 1e-05, atol = 1e-08)

   Check if a quantum state is LDOI (Local Diagonal Orthogonal Invariant).

   A matrix \(A \in \mathcal{L}(\mathcal{X} \otimes \mathcal{Y})\) is called *local diagonal
   orthogonal invariant* (LDOI) if \(\Phi_O(A) = A\), where \(\Phi_O\) is the local
   diagonal orthogonal twirl map defined as:

   \[
       \Phi_O(A) = \frac{1}{2^n} \sum_{O \in \text{DO}(\mathcal{X})} (O \otimes O) A (O \otimes O)
   \]

   where \(\text{DO}(\mathcal{X})\) is the set of diagonal matrices with entries \(\pm 1\).

   LDOI states include many important families such as Werner states, isotropic states, X-states,
   and mixtures of Dicke states. This function efficiently checks the LDOI property using the
   standard basis representation.

   .. rubric:: Examples

   X-states are examples of 2-qubit LDOI states:

   ```python exec="1" source="above"
   from toqito.matrix_props import is_ldoi
   import numpy as np

   # Example X-state
   x_state = np.array([[1, 0, 0, 2],
                        [0, 3, 4, 0],
                        [0, 5, 6, 0],
                        [7, 0, 0, 8]])
   print(is_ldoi(x_state))
   ```

   All diagonal states are LDOI:

   ```python exec="1" source="above"
   from toqito.matrix_props import is_ldoi
   import numpy as np

   diagonal_state = np.diag([1, 2, 3, 4])
   print(is_ldoi(diagonal_state))
   ```

   Non-LDOI states return False:

   ```python exec="1" source="above"
   from toqito.matrix_props import is_ldoi
   import numpy as np

   # Random non-LDOI state
   non_ldoi = np.array([[1, 2, 3, 4],
                         [5, 6, 7, 8],
                         [9, 10, 11, 12],
                         [13, 14, 15, 16]])
   print(is_ldoi(non_ldoi))
   ```

   :param mat: A matrix representing a quantum state.
   :param rtol: Relative tolerance parameter (default: 1e-05).
   :param atol: Absolute tolerance parameter (default: 1e-08).

   :returns: True if the matrix is LDOI, False otherwise.