Operational meaning of the non-repeatability term in the general-measurement bound
Determine the precise operational meaning of the second term in the inequality that upper-bounds |P^{M→N}_ρ(m,n) − P^{α}_ρ(m,n)| for two general quantum measurements M={M_m}_m and N={N_n}_n with POVMs E_M and E_N, namely the quantity ||E_N(n)|| · || M_m(ρ) ∘_α Σ_{m'≠m} E_M(m′) + Σ_{m'≠m} M_{m'}(ρ) ∘_α E_M(m) ||_1, which is tentatively interpreted as evaluating the non-repeatability of M. Ascertain an operational interpretation or experimental procedure that directly corresponds to this term within the framework extending beyond projective measurements.
References
Since M_m(\rho)\circ_{\alpha}\hspace{-7pt}\sum_{m': m'\neq m}\hspace{-7pt}E_M(m')+\hspace{-7pt}\sum_{m': m'\neq m}\hspace{-7pt}M_{m'}(\rho)\circ_{\alpha}E_M(m) can be regarded as the contribution associated with obtaining outcome m' in the post-measurement state conditioned on outcome m, the second term of our bound can be viewed as evaluating the non-repeatability of M. However, as its precise operational meaning remains unclear, further research on the generalization to general measurements is required.