Uncertainty

David Wick

Ming‐Jing Zhao, Yuan-Hong Tao

Abstract The uncertainty relation is a distinctive characteristic of quantum theory. The uncertainty is essentially rooted in quantum states. In this work we regard the uncertainty as an intrinsic property of quantum state and characterize it systematically with respect to given projective measurement. Some basic concepts about uncertainty are reformulated in this context. We prove and get the form of the uncertainty preserving operations. The quantum states with maximal uncertainty are characterized. A universal decomposition of uncertainty into classical uncertainty and quantum uncertainty is provided. Furthermore, a unified and general relation among uncertainty, coherence and coherence of assistance is established. These results are independent of any explicit uncertainty measure. At last, we propose a new uncertainty measure called the geometric uncertainty based on the fidelity and link it with the geometric coherence.

Varun Narasimhachar, Alireza Poostindouz, Gilad Gour

Historically, the element of uncertainty in quantum mechanics has been expressed through mathematical identities called uncertainty relations, a great many of which continue to be discovered. These relations use diverse measures to quantify uncertainty (and joint uncertainty). In this paper we use operational information-theoretic principles to identify the common essence of all such measures, thereby defining measure-independent notions of uncertainty and joint uncertainty. We find that most existing entropic uncertainty relations use measures of joint uncertainty that yield themselves to a small class of operational interpretations. Our notion relaxes this restriction, revealing previously unexplored joint uncertainty measures. To illustrate the utility of our formalism, we derive an uncertainty relation based on one such new measure. We also use our formalism to gain insight into the conditions under which measure-independent uncertainty relations can be found.

Devon Bowen

Xiao Zheng, Shao‐Qiang Ma, Guo‐Feng Zhang +2 more

We unify the product and the sum form variance-based uncertainties to a Schr\{o}dinger-liked uncertainty relation. Some well-known uncertainties in either product or sum form can be recovered by the Schr\{o}dinger-liked uncertainty, i.e., the product form uncertainty is the Schr\{o}dinger-liked uncertainty relation for Hermitian operators and the sum form uncertainty is actually Schr\{o}dinger-liked uncertainty relation for non-Hermitian operators. Thus the Schr\{o}dinger-liked uncertainty relation provides a unified uncertainty relation. Also, the unified uncertainty reveals some quantum properties that the traditional uncertainty relation cannot do, and indicates that the uncertainty relation for the incompatible observables is bounded not only by the commutation relation between themselves, but also by the quantities related with the other operator, which is named as auxiliary operator. The importance of the unified uncertainty, as well as the concept of the auxiliary operator deduced from it, is demonstrated in explaining and fixing the trivial lower bound appearing in the traditional uncertainty relations.

Ivars Peterson