I was reading a paper (link) in which they implement action at a distance using the annihilation operator, and in it they say that "this observation does not imply superluminal signaling because photon annihilation is not a unitary operation, and as such can be realized only probabilistically".
EDIT: A summary of the theoretical part of the paper in a few words: There's an initial state of $|\psi>|{0}>$ entering a Mach-Zehnder inteferometer. After the first beamsplitter, they apply the annihilation operator to one of its output leg (Alice - $\hat{a}_A$). Since it can be written as a linear combination of the input ports, it is proportional to the mode $\hat{a}$ in which $|\psi>$ was introduced. This implies that a local action of the annihilation operator actually affects the total state (unlocal). Then, they say that "the action at a distance of the photon annihilation operator can be made explicit by observing its effect on the mean number of photons in Bob's mode (the other output port - $\hat{a}_B$)". They give the example of $|\psi>$ being a Fock state $|N>$, and once Alice applies $\hat{a}_A$ on the state, the state in mode $\hat{a}$ becomes $|N-1>$, and so the mean number of photons in Bob's mode changes to $\propto|N-1>$. Afterwards, they claim the sentence I'd like to understand.
I'm not really sure what they mean by that and would be happy if someone could explain this statement to me.
Thanks in advance.