Uncertainty relations provide constraints on how well the outcomes of incompatible measurements can be predicted, and as well as being fundamental to our understanding of quantum theory, they have practical applications such as for cryptography and witnessing entanglement. Here we shed new light on the entropic form of these relations, showing that they follow from a few simple properties, including the data-processing inequality. We prove these relations without relying on the exact expression for the entropy, and hence show that a single technique applies to several entropic quantities, including the von Neumann entropy, min- and max-entropies, and the Rényi entropies.
Uncertainty relations provide constraints on how well the outcomes of incompatible measurements can be predicted, and as well as being fundamental to our understanding of quantum theory, they have practical applications such as for cryptography and witnessing entanglement. Here we shed new light on the entropic form of these relations, showing that they follow from a few simple properties, including the data-processing inequality. We prove these relations without relying on the exact expression for the entropy, and hence show that a single technique applies to several entropic quantities, including the von Neumann entropy, min- and max-entropies, and the Rényi entropies.
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 Rényi entropies constitute a family of information measures that generalizes the well-known Shannon entropy, inheriting many of its properties. They appear in the form of unconditional and conditional entropies, relative entropies, or mutual information, and have found many applications in information theory and beyond. Various generalizations of Rényi entropies to the quantum setting have been proposed, most prominently Petz's quasi-entropies and Renner's conditional min-, max-, and collision entropy. However, these quantum extensions are incompatible and thus unsatisfactory. We propose a new quantum generalization of the family of Rényi entropies that contains the von Neumann entropy, min-entropy, collision entropy, and the max-entropy as special cases, thus encompassing most quantum entropies in use today. We show several natural properties for this definition, including data-processing inequalities, a duality relation, and an entropic uncertainty relation.
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 R\'enyi entropies constitute a family of information measures that generalizes the well-known Shannon entropy, inheriting many of its properties. They appear in the form of unconditional and conditional entropies, relative entropies or mutual information, and have found many applications in information theory and beyond. Various generalizations of R\'enyi entropies to the quantum setting have been proposed, most notably Petz's quasi-entropies and Renner's conditional min-, max- and collision entropy. Here, we argue that previous quantum extensions are incompatible and thus unsatisfactory.
We propose a new quantum generalization of the family of R\'enyi entropies and the corresponding relative entropies that contains the min-entropy, collision entropy, von Neumann entropy as well as the max-entropy as special cases, thus encompassing most quantum entropies in use today. We show several natural properties for this definition, including data-processing inequalities and a limited duality relation.
We conclude that the treatment of these entropies is technically challenging and requires sophisticated tools from linear algebra. We share several conjectured properties which we were unable to prove.
Abstract Quantum entanglement is regarded as one of the core concepts, which is used to describe the non-classical correlation between subsystems, and entropic uncertainty relation plays a vital role in quantum precision measurement. It is well known that entanglement of formation can be expressed by von Neumann entropy of subsystems for arbitrary pure states. An interesting question is naturally raised: is there any intrinsic correlation between the entropic uncertainty relation and quantum entanglement? Or if the relation can be applied to estimate the entanglement. In this work, we focus on exploring the complementary relation between quantum entanglement and the entropic uncertainty relation. The results show that there exists an inequality relation between both of them for an arbitrary two-qubit system, and specifically the larger uncertainty will induce the weaker entanglement of the probed system, and vice versa. Besides, we use randomly generated states as illustrations to verify our results. Therefore, we claim that our observations might offer and support the validity of using the entropy uncertainty relation to estimate quantum entanglement.