Generalized uncertainty relations: Existing paradigms and new approaches

Guolin Wu

The classical world, governed by classical physics, might have felt more intuitive and certain. However, the 20th century’s revolutionary birth of quantum mechanics opened a way to the “quantum world”. This chapter investigates the evolving concept of “world”, especially exploring the special and general theories of relativity, with Wittgenstein’s philosophical view that the world is composed of facts, or the existence of states of things. Quantum mechanics has two essential principles: The Heisenberg uncertainty principle and the new Heisenberg uncertainty principle, where the former indicates the inherent uncertainty within the quantum world and the latter uncovers the certainty under some circumstances such as quantum entanglement. Quantum mechanics, while having multiple interpretations, is considered a relatively complete framework; however, its “counterfactuals”—scientific facts that defy classical expectations—can be challenging for the classical understanding. Additionally, the potential for “backward causality” under time symmetry conditions adds another level of complexity. These counterfactuals and the possibility of backward causality don’t contradict existing facts, but rather signal the expansion of our scientific scope and, potentially, a new scientific paradigm.

Burcu Elif Akten

Berthold‐Georg Englert

J. R. Croca

The author of this book presents conceptual and experimental evidence showing that Heisenberg's uncertainty relations are not valid in all cases. Furthermore, he derives a more general set of uncertainty relations. The new relations result from the replacement of the Fourier nonlocal and nontemporal paradigm by wavelet local analysis. These results lead to a coherent and beautiful causal synthesis unifying quantum and classical physics.

Huangjun Zhu

The existence of observables that are incompatible or not jointly measurable is a characteristic feature of quantum mechanics, which lies at the root of a number of nonclassical phenomena, such as uncertainty relations, wave--particle dual behavior, Bell-inequality violation, and contextuality. However, no intuitive criterion is available for determining the compatibility of even two (generalized) observables, despite the overarching importance of this problem and intensive efforts of many researchers. Here we introduce an information theoretic paradigm together with an intuitive geometric picture for decoding incompatible observables, starting from two simple ideas: Every observable can only provide limited information and information is monotonic under data processing. By virtue of quantum estimation theory, we introduce a family of universal criteria for detecting incompatible observables and a natural measure of incompatibility, which are applicable to arbitrary number of arbitrary observables. Based on this framework, we derive a family of universal measurement uncertainty relations, provide a simple information theoretic explanation of quantitative wave--particle duality, and offer new perspectives for understanding Bell nonlocality, contextuality, and quantum precision limit.