Reformulating the Quantum Uncertainty Relation

TL;DR

This paper introduces a new, state-independent quantum uncertainty relation that provides complete trade-offs for observable variances, overcoming limitations of previous relations and offering a geometric perspective on measurement constraints.

Contribution

The work presents a novel form of uncertainty relation that is quantum state independent and addresses the triviality problem, enhancing understanding of measurement limitations in quantum systems.

Findings

Provides a state-independent uncertainty bound for variances

Addresses the triviality problem in existing uncertainty relations

Offers a geometric interpretation of measurement limitations

Abstract

Uncertainty principle is one of the cornerstones of quantum theory. In the literature, there are two types of uncertainty relations, the operator form concerning the variances of physical observables and the entropy form related to entropic quantities. Both these forms are inequalities involving pairwise observables, and are found to be nontrivial to incorporate multiple observables. In this work we introduce a new form of uncertainty relation which may give out complete trade-off relations for variances of observables in pure and mixed quantum systems. Unlike the prevailing uncertainty relations, which are either quantum state dependent or not directly measurable, our bounds for variances of observables are quantum state independent and immune from the "triviality" problem of having zero expectation values. Furthermore, the new uncertainty relation may provide a geometric explanation for the reason why there are limitations on the simultaneous determination of different observables in $N$-dimensional Hilbert space.