toqito.matrix_props.trace_norm
==============================

.. py:module:: toqito.matrix_props.trace_norm

.. autoapi-nested-parse::

   Computes the trace norm metric of a density matrix.





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

.. py:function:: trace_norm(rho)

   Compute the trace norm of the state [@Quantiki_TrNorm].

   Also computes the operator 1-norm when inputting an operator.

   The trace norm \(||\rho||_1\) of a density matrix \(\rho\) is the sum of the singular
   values of \(\rho\). The singular values are the roots of the eigenvalues of
   \(\rho \rho^*\).

   .. rubric:: Examples

   Consider the following Bell state

   \[
       u = \frac{1}{\sqrt{2}} \left( |00 \rangle + |11 \rangle \right) \in \mathcal{X}.
   \]

   The corresponding density matrix of \(u\) may be calculated by:

   \[
       \rho = u u^* = \begin{pmatrix}
                        1 & 0 & 0 & 1 \\
                        0 & 0 & 0 & 0 \\
                        0 & 0 & 0 & 0 \\
                        1 & 0 & 0 & 1
                      \end{pmatrix} \in \text{D}(\mathcal{X}).
   \]

   It can be observed using `|toqito⟩` that \(||\rho||_1 = 1\) as follows.

   ```python exec="1" source="above"
   from toqito.states import bell
   from toqito.matrix_props import trace_norm

   rho = bell(0) @ bell(0).conj().T

   print(trace_norm(rho))
   ```

   :param rho: Density operator.

   :returns: The trace norm of `rho`.