Subadditivity for Renyi entropies Do the Renyi entropies satisfy the standard subadditivity of Shannon entropy? That is, 
\begin{equation}
H_\alpha(A,B) \leq H_\alpha(A) + H_\alpha(B)  ? 
\end{equation}
for $\alpha \ne 1$. If they do, for which $\alpha$? 
Here $H_\alpha(X)$ is the standard Renyi $\alpha$-entropy of the random variable X,
\begin{equation}
H_\alpha(X)=\frac{1}{1-\alpha}\log\sum_i^n p_i^\alpha,
\end{equation}
and $H_\alpha(A,B)$ is the Renyi entropy of the joint probability distribuition of the random variables A and B.
 A: No, the Renyi entropy is not subadditive. It also lacks several other "natural" properties of entropies. 

See this paper on "Additive entropies of degree-$q$ and the Tsallis Entropy by B. H. Lavenda and J. Dunning-Davies for more details, references, and versions of entropy, which possess many desired Shannon-entropy-like properties, while generalizing it.

A: Suvrit has answered it completely, but let me suggest how you might go about finding counterexamples.
It's often useful to work with not the Rényi entropies but their exponentials,
$$
D_\alpha(X) = \exp(H_\alpha(X)) = \Bigl( \sum_{i=1}^n p_i^\alpha \Bigr)^{1/(1-\alpha)}
$$
(where, as in your question, $X$ is a random variable with distribution $p_1, \ldots, p_n$).  One advantage of working with $D$ rather than $H$ is that there's a useful limit as $\alpha \to \infty$, namely
$$
D_\infty(X) = 1/\max_i p_i.
$$
Since this is such a simple formula, $\alpha = \infty$ is a good case to try when testing conjectures.
In terms of $D$, subadditivity becomes $D_\alpha(A, B) \leq D_\alpha(A) D_\alpha(B)$.  It's easy to find counterexamples when $\alpha = \infty$: for instance,
$$
\begin{pmatrix} 
1/2  &1/4 \\\
1/4 &0
\end{pmatrix}
$$
is a counterexample since
$$
\frac{1}{\max\{1/2, 1/4, 1/4, 0\}} = 2 > \frac{16}{9} = \frac{1}{\max\{3/4,1/4\}}\frac{1}{\max\{3/4,1/4\}}.
$$
It follows that this is a counterexample for all sufficiently large finite $\alpha$.
If you graph it, you see that it is in fact a counterexample for all $\alpha$ greater than about $1.6$.  Tweaking it gives you counterexamples for all $\alpha > 1$.
A: In fact, to find a violation of any linear inequality you could form using the Renyi entropies of order $\alpha \in (0,1) \cup (1,\infty)$, please consult the following paper:
http://arxiv.org/abs/1212.0248
