  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.
