Statistical Certification of Majorana fermions

Physicists conjectured the existence of Majorana zero-energy modes in various condensed matter setups. Several experimental groups have even reported evidence of their existence by means that do not reflect or exploit its non-Abelian braiding statistics, a prerequisite for topological computation. What constitutes conclusive evidence for Majorana fermion detection? We present a quantum self-testing protocol that uses minimal assumptions and establishes strong conditions for Majorana fermion detection. We propose a contextuality witness W which consists of putative Majorana fermion parity operators. The resulting Bell-like inequality <W> ≤ 3 can be violated only if the system exhibits quantum contextuality. We further show that observing the ideal measurement statistics, leading to <W>=5, implies anticommutativity of the implemented Majorana fermion parity operators, a necessary prerequisite for the non-Abelian braiding statistics of Majoranas. Our protocol is robust to experimental errors. We obtain lower bounds on the fidelities of the state and measurement operators using both analytical calculations and a Semi Definite Programming approach. We also propose a protocol that certifies the gates which are implemented by the braiding of Majorana modes.