The Optimal Uncertainty Relation

Employing the lattice theory on majorization, the universal quantum uncertainty relation for any number of observables and general measurement is investigated. It is found that 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; and 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 found can be mimicked by the Lorenz curve, initially introduced in economics to measure the wealth concentration degree of a society. The lattice theory and Lorenz curve are found, surprisingly, to provide novel insights into the quantum uncertainty relation. Different from variance and entropy, the majorization lattice induces a relative measure to the quantum uncertainty. The degree of incompatibility of observables can be mimicked by the Lorenz curve which normally represents the degree of wealth concentration in a society.