In the quantum world, the famous Heisenberg uncertainty principle bounds the product of the variances of two incompatible observables such as the position and the momentum of a particle by the Planck constant. Heisenberg, one of the founders of quantum mechanics, gave only an intuitive formulation of this principle, using thought experiments. Later, the uncertainty relation was generalized by Robertson for general observables and it was proved that the product of the variances of two incompatible observables is bounded by their commutator (a quantity that gives the difference between two physical observables when multiplied in succession).
However, the flaw of the Heisenberg relation lies in the fact that due to its product structure, it cannot fully capture the concept of incompatible observables. The lower bound for the product of variances can be zero even for noncommuting observables, making the relation trivial. In the past, there have been other formulations of uncertainty relations in terms of bounds on the sum of information-theoretic quantities that measure the uncertainty of measurement outcomes for two incompatible observables.
However, in the laboratory, the experimentally measured error bars are directly connected to the variance and not to information-theoretic quantities such as the Shannon entropy.
In the present Physical Review Letters paper [L. Maccone and A. K. Pati, Phys. Rev. Lett. (2014)], these authors have proved two new uncertainty relations that bound the sum of variances of two incompatible observables. In contrast to the well-known Heisenberg-Robertson relation, which bounds the product of variances, these new relations always give non-trivial bounds for all incompatible observables.
To read more, click here.