Experimental demonstration of tight duality relation in three-path interferometer**Project supported by the National Key Research and Development Program of China (Grant Nos. 2016YFA0302700 and 2017YFA0304100), the National Natural Science Foundation of China (Grant Nos. 11822408, 11674304, 11774335, 61490711, 11474267, 11821404, 11325419, 11904356, and 91321313), the Youth Innovation Promotion Association, Chinese Academy of Sciences (Grant No. 2017492), the Foundation for Scientific

Abstract

Bohr’s principle of complementarity has a long history and it is an important topic in quantum theory, among which the famous example is the duality relation. The relation between visibility C and distinguishability D, , has long been recognized as the only representative of the duality relation. However, recent researches have shown that this inequality is not good enough because it is not tight for multipath interferometers. Meanwhile, a tight bound for the multipath interferometer has been put forward. Here we design and experimentally implement a three-path interferometer coupling with path indicator states. The wave property of photons is characterized by l1-norm coherence measure, and the particle property is based on distinguishability of the indicator states. The new duality relation of the three-path interferometer is demonstrated in our experiment, which bounds the union of a right triangle and a part of elliptical area inside the quadrant of a unit circle. Data analysis confirms that the new bound is tight for photons in three-path interferometers.