Adding or subtracting a single photon is the same for pure squeezed vacuum states

The addition of a single photon to a light field can lead to exactly the same outcome as the subtraction of a single photon: starting from the same initial state, both procedures can generate the same final quantum state. We prove that this identity-of-outcome is true for pure squeezed vacuum states of light, and in some sense only for those. We show that mixed states can show this identity-of-outcome for the addition or subtraction of a photon if they are generated from the incoherent sums of pure squeezed vacuum states with the same squeezing. We point out that our results give a reinterpretation to the fact that pure squeezed vacuum states, with squeezing e−z, are formally annihilated by Bogoliubov-transformed annihilation operators: âz=âcosh(z)−â†sinh(z).