Variance-based uncertainty relation for incompatible observers

Xiao Hong Zheng, Guo‐Feng Zhang

Bin Chen, Ning-Ping Cao, Shao-Ming Fei +1 more

Yunlong Xiao, Naihuan Jing, Bing Yu +2 more

Uncertainty relations occupy a fundamental position in quantum mechanics. We propose stronger variance-based uncertainty relations for the product and sum of variances of two incompatible observables in a finite dimensional Hilbert space. It is shown that the new uncertainty relations provide near-optimal state-dependent bounds, which can be useful for quantum metrology, entanglement detection etc. in quantum information theory. It is further shown that the uncertainty relations are related to the spreads of the distribution of measurement outcomes caused by incompatible observables. Intuitively, this means that the ability of learning the distribution has both the upper and lower bounds. Combination of these bounds provides naturally an uncertainty interval which captures the essence of uncertainty in quantum theory. Finally, we explain how to employ entropic uncertainty relations to derive lower bounds for the product of variances of incompatible observables.

Lorenzo Maccone, Arun Kumar Pati

The Heisenberg-Robertson uncertainty relation expresses a limitation in the possible preparations of the system by giving a lower bound to the product of the variances of two observables in terms of their commutator. Notably, it does not capture the concept of incompatible observables because it can be trivial; i.e., the lower bound can be null even for two noncompatible observables. Here we give two stronger uncertainty relations, relating to the sum of variances, whose lower bound is guaranteed to be nontrivial whenever the two observables are incompatible on the state of the system.

Lei Xiao, Bowen Fan, Kunkun Wang +2 more

This paper shows experimental tests of state dependent uncertainty relations for the product as well as the sum of variances of two incompatible observables for photonic qutrits. These uncertainty relations are independent of any optimization and much tighter than the Robertson-Schrodinger uncertainty relation and other ones found in current literature. Also the authors test the state dependent reverse uncertainty relations for the sum of variances of two incompatible observables.

Zheng Xu, Qiong Guo