The Optimal Uncertainty Relation

Abstract: Employing the lattice theory on majorization, we investigate the universal quantum uncertainty relation for any number observables and general measurement. We find: 1. The least bounds of the universal uncertainty relations can only be properly defined in the lattice theory; 2. Contrary to variance and entropy, the metric induced by the majorization lattice implies an intrinsic structure of the quantum uncertainty; 3. The lattice theory correlates the optimization of uncertainty relation with the entanglement transformation under local quantum operation and classical communication. Interestingly, the optimality of the universal uncertainty relation is found can be mimicked by the Lorenz curve, initially introduced in economics to measure the wealth concentration degree of a society.