Conservation law for Uncertainty relations and quantum correlations

Uncertainty principle, a fundamental principle in quantum physics, has been studied intensively via various uncertainty inequalities. Here we derive an uncertainty equality in terms of linear entropy, and show that the sum of uncertainty in complementary local bases is equal to a fixed quantity. We also introduce a measure of correlation in a bipartite state, and show that the sum of correlations revealed in a full set of complementary bases is equal to the total correlation in the bipartite state. The surprising simple equality relations we obtain imply that the study on uncertainty principle and correlations can rely on the use of linear entropy, a simple quantity that is very convenient for calculation.