Stronger uncertainty relations

Heisenberg's uncertainty relation is one of the fundamental results in quantum mechanics.^[1] Later Robertson proved the uncertainty relation for two general non-commuting observables,^[2] which was strengthened by Schrödinger.^[3] However, the conventional uncertainty relation like the Robertson-Schrödinger relation cannot give a non-trivial bound for the product of variances of two incompatible observables because the lower bound in the uncertainty inequalities can be null and hence trivial even for observables that are incompatible on the state of the system. The Heisenberg–Robertson–Schrödinger uncertainty relation was proved at the dawn of quantum formalism and is ever-present in the teaching and research on quantum mechanics. After about 85 years of existence of the uncertainty relation this problem was solved recently by Lorenzo Maccone and Arun K. Pati. The standard uncertainty relations are expressed in terms of the product of variances of the measurement results of the observables ${\displaystyle A}$ and ${\displaystyle B}$, and the product can be null even when one of the two variances is different from zero. However, the stronger uncertainty relations due to Maccone and Pati provide different uncertainty relations, based on the sum of variances that are guaranteed to be nontrivial whenever the observables are incompatible on the state of the quantum system.^[4] (Earlier works on uncertainty relations formulated as the sum of variances include, e.g., He et al.,^[5] and Ref.^[6] due to Huang.)