toqito.state_props.l1_norm_coherence
====================================

.. py:module:: toqito.state_props.l1_norm_coherence

.. autoapi-nested-parse::

   Computes the l1-norm of coherence of a quantum state.





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

.. py:function:: l1_norm_coherence(rho)

   Compute the l1-norm of coherence of a quantum state [@Rana_2017_Log].

   The \(\ell_1\)-norm of coherence of a quantum state \(\rho\) is
   defined as

   \[
       C_{\ell_1}(\rho) = \sum_{i \not= j} \left|\rho_{i,j}\right|,
   \]

   where \(\rho_{i,j}\) is the \((i,j)^{th}\)-entry of \(\rho\)
   in the standard basis.

   The \(\ell_1\)-norm of coherence is the sum of the absolute values of
   the sum of the absolute values of the off-diagonal entries of the density
   matrix `rho` in the standard basis.

   This function was adapted from QETLAB.

   .. rubric:: Examples

   The largest possible value of the \(\ell_1\)-norm of coherence on
   \(d\)-dimensional states is \(d-1\), and is attained exactly by
   the "maximally coherent states": pure states whose entries all have the
   same absolute value.

   ```python exec="1" source="above"
   from toqito.state_props import l1_norm_coherence
   import numpy as np
   # Maximally coherent state.
   v = np.ones((3,1))/np.sqrt(3)
   print(l1_norm_coherence(v))
   ```

   :param rho: A matrix or vector.

   :returns: The l1-norm coherence of `rho`.