No-go theorems for photon state transformations in quantum linear optics

We give a necessary condition for photon state transformations in linear optical setups preserving the total number of photons. From an analysis of the algebra describing the quantum evolution, we find a conserved quantity that appears in all allowed optical transformations. We give some examples and numerical applications, with example code, and give three general no-go results. These include (i) the impossibility of deterministic transformations which redistribute the photons from one to two different modes, (ii) a proof that it is impossible to generate a perfect Bell state with an arbitrary ancilla from the Fock basis and (iii) a restriction for the conversion between different types of entanglement (converting GHZ to W states). These tools and results can help in the design of experiments for optical quantum state generation.