How does even Majorana Fermion exist ? Why any fermion can be written as a combination of two Majorana fermions? Is there any physical meaning in it? Why Majorana fermion can be used for topological quantum computation?
The mathematics is clear-Majorana fermions are fermions, that don't carry an ``electric'' charge (it's the fermionic analog of the neutral scalar); the only question is, whether any physical systems exist, where such degrees of freedom can be identified. It seems that such systems have been found, e.g. http://www.2physics.com/2014/12/observation-of-majorana-fermions-in.html
The reason Majorana fermions are useful in describing topological quantum computation is, precisely, because they don't carry an ``electric'' charge, thus are sensitive only to ``topological'' properties.
I studied physics long ago, longer, that most of the authors here live. Yet, I believe that the statement "Why any fermion can be written as a combination of two Majorana fermions?" is incorrect. Majorana fermions are neutral, henceforth, they should obey the symmetry C --> -C, which cannot be violated by any linear combination of their wavefunctions. What is really meant, is that because of 1/2 spin of these fermions, a wave function of a (neutral) fermion can be produced as a combination of S- and P- two-particle wavefunctions. Certainly, the Majorana fermions have been observed in the solid-state context, even on Wikipedia and references thereof.
The wave function of a Majorana fermion describes a one-particle state, not a two-particle state; so it's misleading to mix these up.
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