A bound on expectation values and variances of quantum observable via Rényi entropy and Tsallis entropy

Entropy is a key concept of quantum information theory. The entropy of a quantum system is a measure of its randomness and has many applications in quantum communication protocols, quantum coherence, and so on. In this paper, based on the Rényi entropy and Tsallis entropy, we derive the bounds of the expectation value and variance of quantum observable respectively. By the maximal value of Rényi entropy, we show an upper bound on the product of variance and entropy. Furthermore, we obtain the reverse uncertainty relation for the product and sum of the variances for [Formula: see text] observables respectively.

What is Time–Energy Uncertainty?

Henry scores a surprise win over Eve thanks to his quantum rocket that is powered by a quantum-chargeable battery. This gadget is subject to the time–energy uncertainty relation that may result in the battery having more energy than expected. This occurs if an energy measurement within a short time “collapses” the battery randomly to the highest energy state. Intriguingly, time is not a quantum observable. This raises the question that was hotly debated by Bohr and Einstein: how can time be uncertain and affect the energy uncertainty? The more general question is: what is the meaning of time, energy and their uncertainty in physics and in human experience? Attempts to define time have been the subject of philosophical controversy throughout millennia. The appendix to this chapter introduces the Schrödinger equation that governs the dynamics of quantum systems and their time–energy uncertainty.