Max - Planck - Institut für Mathematik in den Naturwissenschaften Leipzig Sum uncertainty relations for arbitrary N incompatible observables

We formulate uncertainty relations for arbitrary N observables. Two uncertainty inequalities are presented in terms of the sum of variances and standard deviations, respectively. The lower bounds of the corresponding sum uncertainty relations are explicitly derived. These bounds are shown to be tighter than the ones such as derived from the uncertainty inequality for two observables [Phys. Rev. Lett. 113, 260401 (2014)]. Detailed examples are presented to compare among our results with some existing ones. Uncertainty principle, as one of the most fascinating features of the quantum world, has attracted considerable attention since the innovation of quantum mechanics. The corresponding uncertainty inequalities are of great importance for both theoretical investigation and experimental implementation. In fact, the Heisenberg uncertainty principle [1–3] typically said that measuring some observables on a quantum system will inevitably disturb the system. There are many ways to quantify the uncertainty of measurement outcomes, for instance, in terms of the noise and disturbance [4, 5], according to successive measurements [6–9], as informational recourses [10], in entropic terms [11, 12], and by means of majorization technique [13–15]. The traditional approach that deals with quantum uncertainties raised in many different experiments uses the same pre-measurement state. For a pair of observables A and B, the well-known Heisenberg-Robertson uncertainty relation [1, 16] says that, ∆A∆B ≥ 1 2 |⟨ψ|[A,B]|ψ⟩|, (1) where ∆(Ω) = √ ⟨Ω2⟩ − ⟨Ω⟩2 is the standard deviation of an observable Ω, and [A,B] = AB−BA. Heisenberg-Robertson uncertainty relation implies the impossibility to determine