toqito.matrix_props.majorizes
=============================

.. py:module:: toqito.matrix_props.majorizes

.. autoapi-nested-parse::

   Determine if one vector or matrix majorizes another.





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

.. py:function:: majorizes(a_var, b_var)

   Determine if one vector or matrix majorizes another [@WikiMajorization].

   Given \(a, b \in \mathbb{R}^d\), we say that \(a\) **weakly majorizes** (or dominates)
   \(b\) from below if and only if

   \[
       \sum_{i=1}^k a_i^{\downarrow} \geq \sum_{i=1}^k b_i^{\downarrow}
   \]

   for all \(k \in \{1, \ldots, d\}\).

   This function was adapted from the QETLAB package.

   .. rubric:: Examples

   Simple example illustrating that the vector \((3, 0, 0)\) majorizes the vector
   \((1, 1, 1)\).

   ```python exec="1" source="above"
   from toqito.matrix_props import majorizes

   print(majorizes([3, 0, 0], [1, 1, 1]))
   ```


   The majorization criterion says that every separable state
   \(\rho \in \text{D}(\mathcal{A} \otimes \mathcal{B})\) is such that
   \(\text{Tr}_{\mathcal{B}}(\rho)\) majorizes
   \(\text{Tr}_{\mathcal{A}}(\rho)\).

   ```python exec="1" source="above"
   from toqito.matrix_props import majorizes
   from toqito.states import max_entangled
   from toqito.matrix_ops import partial_trace

   v_vec = max_entangled(3)
   rho = v_vec @ v_vec.conj().T

   print(majorizes(partial_trace(rho, [1]), rho))
   ```

   :param a_var: Matrix or vector provided as list or np.array.
   :param b_var: Matrix or vector provided as list or np.array.

   :returns: Return `True` if `a_var` majorizes `b_var` and `False` otherwise.