On Quantum Operations of Photon Subtraction and Photon Addition

Abstract

The conventional photon subtraction and photon addition transformations, ̺ → ta̺a † and ̺ → ta † ̺a, are not valid quantum operations for any constant t > 0 since these transformations are not trace nonincreasing. For a fixed density operator ̺ there exist fair quantum operations, N− and 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 ̺ 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[̺H 2 ] ≤ 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 ̺ such that tr[̺H] ≥ E1 > 0 and tr[̺H 2 ] ≤ E2 < ∞. We prove that these conditions cannot be relaxed and generalize the results to the cases of multiple photon subtraction and addition.

Trace decreasing quantum dynamical maps: Divisibility and entanglement dynamics

“…A quantum operation Λ corresponds to a specific measurement outcome, say, a successful detection of the heralding photon in quantum optics experiments [1,2,4,6]. In mathematical description of experiments with photon subtraction or photon addition [4,6], however, the conventionally used transformations → ta a † and → ta † a are not legitimate quantum operations for any constant t > 0 due to the violation of the trace nonincreasing property [13] (a and a † are the photon annihilation operator and the photon creation operator for a considered mode, respectively). The transformations above approximate the faithful quantum operations of photon substraction and addition for density operators from some class [13].…”