All-order evaluation of weak measurements: --- The cases of an operator   ${\bf A}$ which satisfies the property ${\bf A}^{2}=1$ ---
  Some exact formulae of the expectation values and probability densities in a weak measurement for an operator ${\bf A}$ which satisfies the property ${\bf A}^{2}=1$ are derived. These formulae include all-order effects of the unitary evolution due to the von-Neumann interaction. These are valid not only in the weak measurement regime but also in the strong measurement regime and tell us the connection between these two regime. Using these formulae, arguments of the optimization of the signal amplification and the signal to noise ratio are developed in two typical experimental setups. 
Kouji Nakamura, Atsushi Nishizawa, and Masa-Katsu Fujimoto
quant-ph math-ph math.MP