 Solution

A local observable is parity even by part (a). In the infinite-separation limit, any nontrivial parity-even product of zero-mode Majoranas that acts on the encoded information has support near at least two separated zero modes and cannot occur in a single local observable. The projection of $O$ onto the fixed-parity ground space is consequently scalar. For every orthonormal ground-space basis,This is Local indistinguishability of separated Majorana zero modes.

At finite but large $\ell /\xi$, zero-mode wavefunctions overlap by $O(e^{-\ell /\xi })$. The Hamiltonian acquires couplings $i{ϵ}_{ij}{\gamma }_i{\gamma }_j$, the ground-state degeneracy is split, and local observables gain logical matrix elements of the same exponential order: