On quantum operations of photon subtraction and photon addition

The conventional photon subtraction and photon addition transformations, $varrho rightarrow t a varrho a^{dag}$ and $varrho rightarrow t a^{dag} varrho a$, are not valid quantum operations for any constant $t>0$ since these transformations are not trace nonincreasing. For a fixed density operator $varrho$ there exist fair quantum operations, ${cal N}_{-}$ and ${cal N}_{+}$, whose conditional output states approximate the normalized outputs of former transformations with an arbitrary accuracy. However, the uniform convergence for some classes of density operators $varrho$ has remained essentially unknown. Here we show that, in the case of photon addition operation, the uniform convergence takes place for the energy-second-moment-constrained states such that ${rm tr}[varrho H^2] leq E_2 < infty$, $H = a^{dag}a$. In the case of photon subtraction, the uniform convergence takes place for the energy-second-moment-constrained states with nonvanishing energy, i.e., the states $varrho$ such that ${rm tr}[varrho H] geq E_1 >0$ and ${rm tr}[varrho H^2] leq E_2 < infty$. We prove that these conditions cannot be relaxed and generalize the results to the cases of multiple photon subtraction and addition.