Abstract The uncertainty principle lies at the heart of quantum physics, and is widely thought of as a fundamental limit of the measurement precision of incompatible observables. Here it is shown that the traditional uncertainty relation in fact belongs to the leading order approximation of a generalized uncertainty relation. That is, the leading order linear dependence of observables gives the Heisenberg type of uncertainty relations, while higher order nonlinear dependence may reveal more different and interesting correlation properties. Applications of the generalized uncertainty relation and the high order nonlinear dependence between observables in quantum information science are also discussed.
Abstract The uncertainty principle lies at the heart of quantum physics, and is widely thought of as a fundamental limit of the measurement precision of incompatible observables. Here it is shown that the traditional uncertainty relation in fact belongs to the leading order approximation of a generalized uncertainty relation. That is, the leading order linear dependence of observables gives the Heisenberg type of uncertainty relations, while higher order nonlinear dependence may reveal more different and interesting correlation properties. Applications of the generalized uncertainty relation and the high order nonlinear dependence between observables in quantum information science are also discussed.
The Heisenberg uncertainty principle is one of the characteristics of quantum mechanics. With the vigorous development of quantum information theory, uncertain relations have gradually played an important role in it. In particular, in order to solved the shortcomings of the concept in the initial formulation of the uncertainty principle, we brought entropy into the uncertainty relation, after that, the entropic uncertainty relation has exploited the advantages to the full in various applications. As we all know the entropic uncertainty relation has became the core element of the security analysis of almost all quantum cryptographic protocols. This review mainly introduces development history and latest progress of uncertain relations. After Heisenberg's argument that incompatible measurement results are impossible to predict, many scholars, inspired by this viewpoint, have made further relevant investigations. They combined the quantum correlation between the observable object and its environment, and carried out various generalizations of the uncertainty relation to obtain more general formulas. In addition, it also focuses on the entropy uncertainty relationship and quantum-memory-assisted entropic uncertainty relation, and the dynamic characteristics of uncertainty in some physical systems. Finally, various applications of the entropy uncertainty relationship in the field of quantum information are discussed, from randomnesss to wave-particle duality to quantum key distribution.
Heisenberg's uncertainty principle has been understood to set a limitation on measurements, whereas the long standing mathematical formulation does not allow such an interpretation. Recently, a new relation was found by the present author to give a universally valid relation between noise and disturbance in general quantum measurements, and it has become clear that the new relation plays a role of the first principle to derive various quantum limits on measurement and information processing. Here, we discuss the state-dependent notion of precise measurements of a given observable, and consider a perfect distance such that zero distance implies the perfect correlation. Then, we shall show that even in the perfect error notions both the position measuring noise and the momentum disturbance can be arbitrarily small. We also show that it is possible to generalize the new noise-disturbance uncertainty relation to perfect error distances.
Uncertainty principle, which was first introduced by Werner Heisenberg in 1927, forms a fundamental component of quantum mechanics. A graceful aspect of quantum mechanics is that the uncertainty relations between incompatible observables allow for succinct quan- titative formulations of this revolutionary idea: it is impossible to simultaneously measure two complementary variables of a particle in precision. In particular, information theory offers two basic ways to express the Heisenberg’s principle: variance-based uncertainty relations and entropic uncertainty relations. We first investigate the uncertainty relations based on the sum of variances and derive a family of weighted uncertainty relations to provide an optimal lower bound for all situations. Our work indicates that it seems unreasonable to assume a priori that incompatible observables have equal contribution to the variance-based sum form uncertainty relations. We also study the role of mutually exclusive physical states in the recent work and generalize the variance-based uncertainty relations to mutually exclusive uncertainty relations. Next, we develop a new kind of entanglement detection criteria within the framework of marjorization theory and its matrix representation. By virtue of majorization uncertainty bounds, we are able to construct the entanglement criteria which have advantage over the scalar detect- ing algorithms as they are often stronger and tighter. Furthermore, we explore various expression of entropic uncertainty relations, including sum of Shannon entropies, majorization uncer- tainty relations and uncertainty relations in presence of quantum memory. For entropic uncertainty relations without quantum side information, we provide several tighter bounds for multi-measurements, with some of them also valid for Rényi and Tsallis entropies besides the Shannon entropy. We employ majorization theory and actions of the symmetric group to obtain an admixture bound for entropic uncertainty relations with multi-measurements. Comparisons among existing bounds for multi-measurements are also given. However,classical entropic uncertainty relations assume there has only classical side information. For modern uncertainty relations, those who allowed for non-trivial amount of quantum side information, their bounds have been strengthened by our recent result for both two and multi- measurements. Finally, we propose an approach which can extend all uncertainty relations on Shannon entropies to allow for quantum side information and discuss the applications of our entropic framework. Combined with our uniform entanglement frames, it is possible to detect entanglement via entropic uncertainty relations even if there is no quantum side in- formation. With the rising of quantum information theory, uncertainty relations have been established as important tools for a wide range of applications, such as quantum cryptography, quantum key distribution, entanglement detection, quantum metrology, quantum speed limit and so on. It is thus necessary to focus on the study of uncertainty relations.
Heisenberg's uncertainty principle has been understood to set a limitation on measurements; however, the long standing mathematical formulation established by Heisenberg, Kennard, and Robertson does not allow such an interpretation. Recently, a new relation was found to give a universally valid relation between noise and disturbance in general quantum measurements, and it has become clear that the new relation plays a role of the first principle to derive various quantum limits on measurement and information processing in a unified treatment. This paper examines the above development on the noise-disturbance uncertainty principle in the model independent approach based on measurement operator formalism, which is widely accepted to describe a class of generalized measurements in the field of quantum information. We obtain explicit formulas for noise and disturbance of measurements given by measurement operators, and show that projective measurements do not satisfy the Heisenberg-type noise-disturbance relation that is typical in the gamma-ray microscope thought experiments. We also show that the disturbance on a Pauli operator of a projective measurement of another Pauli operator constantly equals to the square root of 2, and examine how this measurement violates the Heisenberg-type relation but satisfies the new noise-disturbance relation.
Unsolved controversies about uncertainty relations and quantum measurements still persists nowadays. They originate around the shortcomings regarding the conventional interpretation of uncertainty relations. Here we show that the respective shortcomings disclose veridic and unavoidable facts which require the abandonment of the mentioned interpretation. So the primitive uncertainty relations appear as being either thought fictions or fluctuations formulae. Subsequently we reveal that the conventional approaches of quantum measurements are grounded on incorrect premises. We propose a new approach in which : (i) the quantum observables are considered as generalized stochastic variables, (ii) the view is focused only on the pre-existent state of the measured system, without any interest for the collapse of the respective state, (iii) a measurement is described as an input-output transformation which modify the probability density and current but preserve the expressions of the operators. The measuring uncertainties are evaluated as changes in the probabilistic estimators of observables. Related to different observables we do not find reasons of principle neither for uncertainties-connections nor for measuring compatibility/incompatibility.
Uncertainty relations involving incompatible observables are one of the cornerstones of quantum mechanics. Aside from their fundamental significance, they play an important role in practical applications, such as detection of quantum correlations and security requirements in quantum cryptography. In continuous variable systems, the spectra of the relevant observables form a continuum and this necessitates the coarse graining of measurements. However, these coarse-grained observables do not necessarily obey the same uncertainty relations as the original ones, a fact that can lead to false results when considering applications. That is, one cannot naively replace the original observables in the uncertainty relation for the coarse-grained observables and expect consistent results. As such, several uncertainty relations that are specifically designed for coarse-grained observables have been developed. In recognition of the 90th anniversary of the seminal Heisenberg uncertainty relation, celebrated last year, and all the subsequent work since then, here we give a review of the state of the art of coarse-grained uncertainty relations in continuous variable quantum systems, as well as their applications to fundamental quantum physics and quantum information tasks. Our review is meant to be balanced in its content, since both theoretical considerations and experimental perspectives are put on an equal footing.