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We propose and analyze a multi-functional setup consisting of high finesseoptical cavities, beam splitters, and phase shifters. The basic scheme projectsarbitrary photonic two-mode input states onto the subspace spanned by theproduct of Fock states |n>|n> with n=0,1,2,.... This protocol does not onlyprovide the possibility to conditionally generate highly entangled photonnumber states as resource for quantum information protocols but also allows oneto test and hence purify this type of quantum states in a communicationscenario, which is of great practical importance. The scheme is especiallyattractive as a generalization to many modes allows for distribution andpurification of entanglement in networks. In an alternative working mode, thesetup allows of quantum non demolition number resolved photodetection in theoptical domain.

Quantum state engineering, purification, and number resolved photon detection with high finesse optical cavities