(Sub-systemic) quantum Zeno effect and single photon interference at a beam splitter

In this work we consider quantum dynamical interaction of single photon with beam splitter after which well-known superposition between reflected and transmitted photon appears. Later reflected photon is absorbed by one and transmitted photon is absorbed by other, distant of two two-level systems of the same type (prepared initially in ground states). It implies entanglement between mentioned two two-level systems. Any two-level system in excited state represents an unstable system that quantum dynamically evolves in a superposition of the initial excited (non-decayed) and ground (decayed) state. We consider frequent detection at only one of two two-level systems and we prove that here a modification of the usual quantum Zeno effect called sub-systemic quantum Zeno effect appears. Namely after mentioned frequent detection two two-level systems are in a specific entangled state with two terms. First term (term of the "frozen" evolution) includes detected two-level system in initial excited state and non-detected two-level system in the ground state. Second term (term of free evolution) includes detected system in ground state and non-detected system in free evolved excited state.