Abstract Besides the well-known Shannon entropy, there is a set of Shannon-like entropies which have applications in statistical and quantum physics. These entropies are functions of certain parameters and converge toward Shannon entropy when these parameters approach the value 1. We describe briefly the most important Shannon-like entropies and present their graphical representations. Their graphs look almost identical, though by superimposing them it appears that they are distinct and characteristic of each Shannon-like entropy. We try to formulate the alternative entropic uncertainty relations by means of the Shannon-like entropies and show that all of them equally well express the uncertainty principle of quantum physics.
Abstract Besides the well-known Shannon entropy, there is a set of Shannon-like entropies which have applications in statistical and quantum physics. These entropies are functions of certain parameters and converge toward Shannon entropy when these parameters approach the value 1. We describe briefly the most important Shannon-like entropies and present their graphical representations. Their graphs look almost identical, though by superimposing them it appears that they are distinct and characteristic of each Shannon-like entropy. We try to formulate the alternative entropic uncertainty relations by means of the Shannon-like entropies and show that all of them equally well express the uncertainty principle of quantum physics.
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.
Abstract Entropic uncertainty relations are studied from the conceptual viewpoint and in the context of various applications. Providing as tight inequalities as possible is thus an important issue. Measurements of special types are indispensable in quantum information science. Their structure allows us to estimate the index of coincidence. Uncertainty relations follow an estimation of entropies at the given index of coincidence. Relations of such a kind are a known issue in information theory. It concerns links between different information measures. Information diagrams are a tool to study relations between two utilized characteristics. A search for improved inequalities uses the index of coincidence as the abscissa and the entropy of interest as the ordinate. This approach before dealt with standard information functions assigned to the Shannon entropy. But generalized entropies often reveal new links between information measures. This paper applies the method of information diagrams to the Rényi and Tsallis entropies. Such entropies and related information functions have found use in quantum information theory. Improved estimates of the Tsallis and Rényi entropies at the given index of coincidence are obtained. Hence, new entropic uncertainty relations for specific types of measurements follow. They include mutually unbiased bases, symmetric informationally complete measurements, and equiangular tight frames.
The Heisenberg uncertainty principle has a more precise formulation in terms of inequalities involving quantum entropies. Currently known entropic uncertainty relations are presented; they capture and extend Heisenberg's idea of the unpredictability of the outcomes of incompatible measurements. Distinct results are obtained for finite- and infinite-dimensional Hilbert spaces. Applications are surveyed, including the formulation of entanglement witnesses, current ideas about wave-particle duality, and the analysis of quantum cryptography.
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.
Abstract We show that a proper expression of the uncertainty relation for a pair of canonically-conjugate continuous variables relies on entropy power, a standard notion in Shannon information theory for real-valued signals. The resulting entropy-power uncertainty relation is equivalent to the entropic formulation of the uncertainty relation due to Bialynicki-Birula and Mycielski, but can be further extended to rotated variables. Hence, based on a reasonable assumption, we give a partial proof of a tighter form of the entropy-power uncertainty relation taking correlations into account and provide extensive numerical evidence of its validity. Interestingly, it implies the generalized (rotation-invariant) Schrödinger–Robertson uncertainty relation exactly as the original entropy-power uncertainty relation implies Heisenberg relation. It is saturated for all Gaussian pure states, in contrast with hitherto known entropic formulations of the uncertainty principle.
Uncertainty is a fundamental characteristic of quantum system. The degree of uncertainty of an observable has long been investigated by the standard deviation of the observable. In recent years, however, by analyzing some special examples, researchers have found that the Shannon entropy of the measurement outcomes of an observable is more suitable to quantify its uncertainty. Formally, Shannon entropy is a special limit of a more general Rényi entropy. In this paper, we discuss the problem of how to predict the measurement outcome of an observable by the existing measurement results of the observable, and how to quantitatively describe the uncertainty of the observable from the perspective of the repeatable probability of the measurement results of this observable in an unknown state. We will argue that if the same observable of different systems in the same state is repeatedly and independently measured many times, then the probability of obtaining an identical measurement result is a decaying function of the number of measurements of obtaining the same result, and the decay rate of the repeatable probability for obtaining the same measurement results and the repeatable number of measurements can represent the degree of uncertainty of the observable in this state. It means that the greater the uncertainty of an observable, the faster the repeatable probability decays with the number of repeatable measurements; conversely, the smaller the uncertainty, the slower the repeatable probability decays with the number of repeatable measurements. This observation enables us to give the Shannon entropy and the Rényi entropy of an observable uniformly by the functional relation between the repeatable probability and the number of repeatable measurements. We show that the Shannon entropy and the Rényi entropy can be formally regarded as the “decay index” of the repeatable probability with the number of repeatable measurements. In this way we also define a generalized Rényi entropy by the repeatable probability for consecutively observing identical results of an observable, and therefore we give a proof of the Maassen-Uffink type entropic uncertainty relation by using this generalized Rényi entropy. This method of defining entropy shows that entropic uncertainty relation is a quantitative limitation for the decay rate of the total probability for obtaining identical measurement results when we simultaneously measure two observables many times.