On Quantum Operations of Photon Subtraction and Photon Addition

Abstract

The conventional photon subtraction and photon addition transformations, ϱ → taϱa^† and ϱ → ta^†ϱa, are not valid quantum operations for any constant t > 0 since these transformations are not trace nonincreasing. For a fixed density operator ϱ there exist fair quantum operations, \(\mathcal{N}_{-}\) and \(\mathcal{N}_{+}\), 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 ϱ 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 tr[ϱH²] ≤ E₂ < ∞, H = a^†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 ϱ such that tr[ϱH] ≥ E₁ > 0 and tr[ϱH²] ≥ E₂ < ∞. We prove that these conditions cannot be relaxed and generalize the results to the cases of multiple photon subtraction and addition.