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We establish a rigorous quantitative connection between (i) theinterferometric duality relation for which-way information and fringevisibility and (ii) Heisenberg's uncertainty relation for position and modularmomentum. We apply our theory to atom interferometry, wherein spontaneouslyemitted photons provide which way information, and unambiguously resolve thechallenge posed by the metamaterial `perfect lens' to complementarity and tothe Heisenberg-Bohr interpretation of the Heisenberg microscope thoughtexperiment.

Complementarity and uncertainty relations for matter wave interferometry