Uncertainty, joint uncertainty, and the quantum uncertainty principle

Abstract

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.

arXiv commentPublished version with major revisions; 8 pages main + bibliography + 1 page appendix; 3 figures

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• Entropic Uncertainty Relations and their Applications
• Sum uncertainty relations based on Wigner-Yanase skew information
• The Conditional Uncertainty Principle
• Entropic uncertainty relations for successive measurements of canonically conjugate observables
• Majorization uncertainty relations for mixed quantum states
• Improved quantum entropic uncertainty relations
• Strong unitary uncertainty relations
• A note on uncertainty relations of arbitrary N quantum channels
• Rényi formulation of uncertainty relations for POVMs assigned to a quantum design
• Operational foundations of complementarity and uncertainty relations
• Tighter sum uncertainty relations via variance and Wigner-Yanase skew information for N incompatible observables
• Quasi-Fine-Grained Uncertainty Relations
• Quantum Uncertainty Equalities and Inequalities for Unitary Operators
• A Universal Quantum Certainty Relation for Arbitrary Number of Observables
• Information and majorization theory for fermionic phase-space distributions