toqito.matrix_props.is_absolutely_k_incoherent
==============================================

.. py:module:: toqito.matrix_props.is_absolutely_k_incoherent

.. autoapi-nested-parse::

   Checks if the matrix is absolutely $k$-incoherent.





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

.. py:function:: is_absolutely_k_incoherent(mat, k, tol = 1e-15)

   Determine whether a quantum state is absolutely k-incoherent [@Johnston_2022_Absolutely].

   Formally, for positive integers \(n\) and \(k\), a mixed quantum state is said to be absolutely k-incoherent
   if \(U \rho U^* \in \mathbb{I}_{k, n}\) for all unitary matrices \(U \in \text{U}(\mathbb{C}^n)\).

   This function checks if the provided density matrix is absolutely k-incoherent based on the criteria introduced in
   [@Johnston_2022_Absolutely] and the corresponding QETLAB functionality [@QETLAB_link]. When
   necessary, an SDP is set up via ``cvxpy``.

   The notion of absolute k-incoherence is connected to the notion of quantum state antidistinguishability as discussed
   in [@Johnston_2025_Tight].

   .. rubric:: Examples

   ```python exec="1" source="above"
   import numpy as np
   from toqito.matrix_props import is_absolutely_k_incoherent
   mat = np.array([[2, 1, 2],
               [1, 2, -1],
               [2, -1, 5]])
   print(is_absolutely_k_incoherent(mat, 4))
   ```

   !!! See
       [is_antidistinguishable()][toqito.state_props.is_antidistinguishable.is_antidistinguishable],
       [is_k_incoherent()][toqito.matrix_props.is_k_incoherent.is_k_incoherent]

   :raises ValueError: If the input matrix is not square.

   :param mat: Matrix to check for absolute k-incoherence.
   :param k: The positive integer indicating the absolute coherence level.
   :param tol: Tolerance for numerical comparisons (default is 1e-15).

   :returns: True if the quantum state is absolutely k-incoherent, False otherwise.