# On quantum operations of photon subtraction and photon addition

## TL;DR

This paper investigates the validity of photon subtraction and addition as quantum operations, proposing approximate operations that converge uniformly under specific energy constraints, and extends these results to multiple photon processes.

## Contribution

It introduces fair quantum operations approximating photon subtraction and addition, establishing conditions for uniform convergence based on energy constraints, and generalizes to multiple photon processes.

## Key findings

- Uniform convergence for photon addition with energy-second-moment constraint.
- Uniform convergence for photon subtraction with energy and non-vanishing energy constraints.
- Conditions for convergence cannot be relaxed.

## Abstract

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.