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$2M$ Majorana zero modes define $M$ complex fermionic modes. Fixing total fermion parity leaves a $2^{M-1}$-dimensional ground space, encoding $M-1$ qubits.

Fermion parity is defined byIt anticommutes with $Q_{\alpha }$ and ${\bar{Q}}_{\dot{\alpha }}$. At fixed nonzero energy and momentum, choose a supercharge combination $S$ for which $\{S,S^{†}\}=cI$ with $c>0$. Taking the finite-dimensional trace over one supermultiplet givesbecause cyclicity of the trace and anticommutation of $(-1{)}^F$ with $S$ make the two terms cancel. ThereforeThis is boson-fermion degeneracy in a supermultiplet.

If explicit or soft supersymmetry breaking terms are added, the supercharge is no longer a conserved symmetry of the full Hamiltonian and states need not form representations of the supersymmetry algebra at equal energy. The positive anticommutator cannot be replaced by a constant on a purported multiplet, so the supertrace proof and mass degeneracy fail.

Fermion parity is $P_f=(-1{)}^F$. An operator $O_{+}$ is parity even when $P_f{O}_{+}{P}_f=O_{+}$, equivalently $[P_f,O_{+}]=0$; it is parity odd when $P_f{O}_{-}{P}_f=-O_{-}$, equivalently $\{P_f,O_{-}\}=0$.

Odd operators supported in disjoint spacelike regions anticommute by the canonical anticommutation relations. If each were an observable, their measurement algebras would fail to commute, allowing the order of spacelike separated measurements to affect predictions. Products of an even number of fermionic fields instead commute at spacelike separation. Locality and compatible spacelike measurements therefore require every physical local observable to be fermion-parity even; parity-odd fields can create charged states but are not themselves observables.