toqito.state_props.is_product
=============================

.. py:module:: toqito.state_props.is_product

.. autoapi-nested-parse::

   Checks if a quantum state is product state.





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

.. py:function:: is_product(rho, dim = None)

   Determine if a given vector is a product state [@WikiSepSt].

   If the input is deemed to be product, then the product decomposition is also
   returned.

   .. 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:

   \[
       \rho = u u^* = \frac{1}{2} \begin{pmatrix}
                        1 & 0 & 0 & 1 \\
                        0 & 0 & 0 & 0 \\
                        0 & 0 & 0 & 0 \\
                        1 & 0 & 0 & 1
                      \end{pmatrix} \in \text{D}(\mathcal{X}).
   \]

   We can provide the input as either the vector \(u\) or the denisty matrix \(\rho\).
   In either case, this represents an entangled state (and hence a non-product state).

   ```python exec="1" source="above" session="is_product_example"
   from toqito.state_props import is_product
   from toqito.states import bell
   rho = bell(0) @ bell(0).conj().T
   u_vec = bell(0)
   print(is_product(rho))
   ```

   ```python exec="1" source="above" session="is_product_example"
   print(is_product(u_vec))
   ```

   :param rho: The vector or matrix to check.
   :param dim: The dimension of the input.

   :returns: `True` if `rho` is a product vector and `False` otherwise.