Uncertainty relations based on information theory for both discrete and continuous distribution functions are briefly reviewed. We extend these results to account for (differential) Rényi entropy and its related entropy power. This allows us to find a new class of information-theoretic uncertainty relations (ITURs). The potency of such uncertainty relations in quantum mechanics is illustrated with a simple two-energy-level model where they outperform both the usual Robertson–Schrödinger uncertainty relation and Shannon entropy based uncertainty relation. In the continuous case the ensuing entropy power uncertainty relations are discussed in the context of heavy tailed wave functions and Schrödinger cat states. Again, improvement over both the Robertson–Schrödinger uncertainty principle and Shannon ITUR is demonstrated in these cases. Further salient issues such as the proof of a generalized entropy power inequality and a geometric picture of information-theoretic uncertainty relations are also discussed.
The uncertainty principle is one of the characteristic properties of quantum theory based on incompatibility. Apart from the incompatible relation of quantum states, mutually exclusiveness is another remarkable phenomenon in the information- theoretic foundation of quantum theory. We investigate the role of mutual exclusive physical states in the recent work of stronger uncertainty relations for all incompatible observables by Mccone and Pati and generalize the weighted uncertainty relation to the product form as well as their multi-observable analogues. The new bounds capture both incompatibility and mutually exclusiveness, and are tighter compared with the existing bounds.
The uncertainty principle is one of the central concepts in quantum theory. Different forms of this principle have been discussed in various foundational and information theoretic topics. Whereas in the discrete input-output scenario the limited nonlocal behavior of quantum theory has been explained by the fine-grained uncertainty relation, in the continuous-variable paradigm the Robertson-Schr\"odinger (RS) uncertainty relation has been used to detect multimode entanglement. Here we show that the RS uncertainty relation plays an important role in discriminating between quantum and postquantum nonlocal correlations in the multimode continuous outcome scenario. We provide a class of
m-mode postquantum nonlocal correlations with a continuous outcome spectrum. While nonlocality of the introduced class of correlations is established through the Calvalcanti-Foster-Reid-Drummond class of Bell inequalities, the RS uncertainty relation detects their postquantum nature. Our result suggests a wider role of the uncertainty principle in the study of nonlocality in continuous-variable multimode systems.