The uncertainty constants: A unified framework of two, three and four observables

TL;DR

The paper develops a unified, matrix-theory based framework for sum-form uncertainty constants across two, three, and four quantum observables. By constructing an operator R (for example $R=H_1+iH_2\,iH_3$ and its four-observable generalization) and leveraging the positivity of $RR^\dagger$, it derives tight lower bounds on the sum of variances, yielding $\sum_{j=1}^3 \Delta H_j^2 \ge \frac{1}{\sqrt{3}} \sum_{j=1}^3 |\langle [H_j,H_{j+1}]\rangle|$ and a four-observable analogue $\sum_{j=1}^4 \Delta H_j^2 \ge \frac{1}{\sqrt{3}} \sum_{ijkl} |\langle [H_i,H_j]\rangle-(-1)^{\tau_{ijkl}}\langle [H_k,H_l]\rangle|$. The approach also recovers Robertson’s relation in the two-observable limit and offers a unified interpretation via Pauli-type structures and quaternions, while simplifying prior proofs. These results provide a compact, generalizable method for uncertainty relations with potential impact on quantum information and foundational studies.

Abstract

Uncertainty is a fundamental and important concept in quantum mechanics. Recent works have revealed both the product and sum forms of uncertainty constants for three observables. Such a result is intimately to the properties of Pauli operators. In this work, using the technique in matrix theory, we give an alternative proof for the case of three observables, and generalize the result to the case of four measurements. Comparing with the original proof, such a derivation is simplified. Moreover, the discussions can deal with the summation form of uncertainty relation for two, three and four observables in a unified way.