Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 342 / 3 / c / Solution

For $a\ne b$, the operator $Q_{ab}=i{\gamma }_a{\gamma }_b$ is Hermitian andThereforeFor $f=({\gamma }_a+i{\gamma }_b)/2$, one has $Q_{ab}=2f^{†}f-1$, so these projectors distinguish the two occupations, equivalently the two values of pair fermion parity. This proves thatdescribe a fermion-parity measurement.

Let $P={\Pi }_{+}^{(N0)}$ and suppose $P|\psi ⟩=|\psi ⟩$. The bilinears $i{\gamma }_a{\gamma }_0$ and $i{\gamma }_b{\gamma }_0$ each anticommute with $i{\gamma }_N{\gamma }_0$, whereas their product commutes with it. Expanding the projectors and sandwiching by $P$ therefore givesAfter normalizing the post-measurement state, the sequence consequently implementson the encoded ground space. This is Measurement-only Majorana braiding.