Comment on "Weak measurements with orbital-angular-momentum pointer states".

In their recent Letter[1], Puentes et al. have provided a scheme for extracting the real and imaginary parts of higher-order weak value by using orbital angular momentum(OAM) states as pointer states, based on two interaction Hamiltonians and multiple joint pointer displacements. They claim that such weak values are inaccessible with Gaussian pointer state only, due to its particular symmetry property. As explained in [1], OAM states introduce a ‘different symmetry’ for the expectation values of operators, and thus provide access to such weak values. The purpose of this Comment is to show that for the same Hamiltonian, Gaussian pointer state by itself can provide access to the real part of higher-order weak value, if suitable pointer displacement is observed. This is because the symmetry property responsible for revealing such value can also be restored with Gaussian pointer state for a suitable choice of the measurement of the observable after post-selection. Puentes et al.[1] considered a joint weak measurement of two commuting observables  and B̂ satisfying (Â) 6= I and (B̂) 6= I. The system-apparatus state at t = 0 is |Ψi〉 = |ψi〉|i〉 = ∫ dxdyψ(x, y)|x, y〉|i〉 where |i〉 denotes pre-selected system state, and the pointer wave function ψ(x, y) is taken to be Laguerre-Gauss mode