It's a very interesting result, but as always with neural networks we have to keep in mind that what matters is not whether a model can be encoded in a different architecture (even at equal entropy), the question is whether the model can be learned in the first place. When you work with shallow nets trained with regular backprop + dropout, you see that their learning capabilities tend to "saturate" much quicker than deep nets. Often with shallow nets, after a point you don't get better results by adding more units or more training data. But deep nets are better able to make use of these extra parameters (extra layers) and extra training data.

Possibly because deep nets conceptually "break down" a learning problem into incremental steps (each new layer being a higher level of representation).

But then again, maybe the problem is simply that we don't have sufficiently good methods for training shallow NNs on large-scale problems. After all, it's only recently that we figured out how to efficiently train deep nets (either pre-training with Autoencoders or RBMs, or through Hessian-free optimization).