toqito.state_props.renyi_entropy
================================

.. py:module:: toqito.state_props.renyi_entropy

.. autoapi-nested-parse::

   Calculates the Rényi entropy metric of a quantum state.





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

.. py:function:: renyi_entropy(rho, alpha)

   Compute the Rényi entropy of a density matrix [@Muller_2013_Renyi_Generalization].

   Let \(P \in \text{Pos}(\mathcal{X})\) be a positive semidefinite operator, for a complex
   Euclidean space \(\mathcal{X}\). Then one defines the *Rényi entropy of order*
   \(\alpha\geqslant0\) as

   \[
       H_{\alpha}(P) = H_{\alpha}(\lambda(P)),
   \]

   where \(\lambda(P)\) is the vector of eigenvalues of \(P\) and where the function
   \(H(\cdot)\) is the classical Rényi entropy of order \(\alpha\) defined as

   \[
       H_{\alpha}(u) = \frac{1}{1-\alpha}\log\left(\sum_{a \in \Sigma} u(a)^{\alpha}\right),
   \]

   where the \(\log\) function is assumed to be the base-2 logarithm, and where
   \(\Sigma\) is an alphabet where \(u \in [0, \infty)^{\Sigma}\) is a vector of
   nonnegative real numbers indexed by \(\Sigma\). It recovers the von Neumann entropy for
   \(\alpha=1\) and the min-entropy for \(\alpha=+\infty\).

   .. 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:

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

   Calculating the Rényi entropy of order \(2\) of \(\rho\) in `|toqito⟩` can be
   done as follows.

   ```python exec="1" source="above"
   from toqito.state_props import renyi_entropy
   import numpy as np
   test_input_mat = np.array(
           [[1 / 2, 0, 0, 1 / 2], [0, 0, 0, 0],
           [0, 0, 0, 0], [1 / 2, 0, 0, 1 / 2]]
       )
   print(renyi_entropy(test_input_mat, 2))
   ```

   Consider the density operator corresponding to the maximally mixed state of dimension two

   \[
       \rho = \frac{1}{2}
       \begin{pmatrix}
           1 & 0 \\
           0 & 1
       \end{pmatrix}.
   \]

   As this state is maximally mixed, the Rényi entropy of \(\rho\) is
   equal to one for all orders \(\alpha\). We can see this in `|toqito⟩` as follows.

   ```python exec="1" source="above"
   from toqito.state_props import renyi_entropy
   import numpy as np
   rho = 1/2 * np.identity(2)
   print(renyi_entropy(rho, 3/2))
   ```

   :param rho: Density operator.
   :param alpha: Order for the Rényi entropy. Note that numerical instability may happen for small positive values because
   :param of the computation of the spectral decomposition.:

   :returns: The Rényi entropy of order `alpha` of `rho`.