The uncertainty principle bounds the uncertainties about incompatible measurements, clearly setting quantum theory apart from the classical world. Its mathematical formulation via uncertainty relations, plays an irreplaceable role in quantum technologies. However, neither the uncertainty principle nor uncertainty relations can fully describe the complementarity between quantum measurements. As an attempt to advance the efforts of complementarity in quantum theories, we formally propose a complementary information principle, significantly extending the one introduced by Heisenberg. First, we build a framework of black box testing consisting of pre- and post-testing with two incompatible measurements, introducing a rigorous mathematical expression of complementarity with definite information causality. Second, we provide majorization lower and upper bounds for the complementary information by utilizing the tool of semidefinite programming. In particular, we prove that our bounds are optimal under majorization due to the completeness of the majorization lattice. Finally, as applications to our framework, we present a general method to outer-approximating all uncertainty regions and also establish fundamental limits for all qualified joint uncertainties.

Yunlong Xiao received his first Ph.D. degree in Mathematical Physics at the Max Planck Institute for Mathematics in the Sciences (MiS) in Leipzig, Germany (under the supervision of Prof. Jürgen Jost and Prof. Naihuan Jing) in February 2017 and his second Ph.D. degree in Pure Mathematics from South China University of Technology (under the supervision of Prof. Naihuan Jing) in June 2017. Following that, he was a Postdoctoral Fellow at the Institute for Quantum Science and Technology at the University of Calgary, Canada (under the supervision of Prof. Barry C. Sanders and Prof. Gilad Gour) until August 2019. He is currently with the Complexity Institute at the School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore as a Research Fellow.