Complementarity relations between various characterizations of a probability distribution are at the core of information theory. In particular, lower and upper bounds for the entropic function are of great importance. In applied topics, we often deal with situations where the sums of certain powers of probabilities are known. The main question is how to convert the imposed restrictions into two-sided estimates on the Shannon entropy. It is addressed in two different ways. The more intuitive of them is based on truncated expansions of the Taylor type. Another method is based on the use of coefficients of the shifted Chebyshev polynomials. We propose here a family of polynomials for estimating the Shannon entropy from below. As a result, estimates are more uniform in the sense that errors do not become too large in particular points. The presented method is used for deriving uncertainty and certainty relations for positive operator-valued measures assigned to a quantum design. Quantum designs are currently the subject of active researches due to potential use in quantum information science. It is shown that the derived estimates are applicable in quantum tomography and detecting steerability of quantum states.
Complementarity relations between various characterizations of a probability distribution are at the core of information theory. In particular, lower and upper bounds for the entropic function are of great importance. In applied topics, we often deal with situations where the sums of certain powers of probabilities are known. The main question is how to convert the imposed restrictions into two-sided estimates on the Shannon entropy. It is addressed in two different ways. The more intuitive of them is based on truncated expansions of the Taylor type. Another method is based on the use of coefficients of the shifted Chebyshev polynomials. We propose here a family of polynomials for estimating the Shannon entropy from below. As a result, estimates are more uniform in the sense that errors do not become too large in particular points. The presented method is used for deriving uncertainty and certainty relations for positive operator-valued measures assigned to a quantum design. Quantum designs are currently the subject of active researches due to potential use in quantum information science. It is shown that the derived estimates are applicable in quantum tomography and detecting steerability of quantum states.
Abstract Information entropies provide powerful and flexible way to express restrictions imposed by the uncertainty principle. This approach seems to be very suitable in application to problems of quantum information theory. It is typical that questions of such a kind involve measurements having one or another specific structure. The latter often allows us to improve entropic bounds that follow from uncertainty relations of sufficiently general scope. Quantum designs have found use in many issues of quantum information theory, whence uncertainty relations for related measurements are of interest. In this paper, we obtain uncertainty relations in terms of min-entropies and Rényi entropies for POVMs assigned to a quantum design. Relations of the Landau–Pollak type are addressed as well. Using examples of quantum designs in two dimensions, the obtained lower bounds are then compared with the previous ones. An impact on entropic steering inequalities is briefly discussed.
Abstract The concept of quantum coherence and its possible use as a resource are currently the subject of active research. Uncertainty and complementarity relations for quantum coherence allow one to study its changes with respect to other characteristics of the process of interest. Protocols of quantum information processing often use measurements that have a special inner structure. Quantum designs are considered as tools with a perspective of fruitful applications in quantum information processing. We obtain uncertainty and certainty relations for coherence averaged with respect to a set of design-structured POVMs of rank one. To characterize the degree of coherence, the relative entropy of coherence is utilized. The derived relations are exemplified with quantum designs in the two-dimensional Hilbert space.
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 Uncertainty relations and quantum entanglement are pivotal concepts in quantum theory. Beyond their fundamental significance in shaping our understanding of the quantum world, they also underpin crucial applications in quantum information theory. In this article, we investigate entropic uncertainty relations and entanglement detection with an emphasis on quantum measurements with design structures. On the one hand, we derive improved Rényi entropic uncertainty relations for design-structured measurements, exploiting the property that the sum of powered (e.g. squared) probabilities of obtaining different measurement outcomes is now invariant under unitary transformations of the measured system and can be easily computed. On the other hand, the above property essentially imposes a state-independent upper bound, which is achieved at all pure states, on one’s ability to predict local outcomes when performing a set of design-structured measurements on quantum systems. Realizing this, we also obtain criteria for detecting multipartite entanglement with design-structured measurements.
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 concept of quantum coherence, including various ways to quantify the degree of coherence with respect to the prescribed basis, is currently the subject of active research. The complementarity of quantum coherence in different bases was studied by deriving upper bounds on the sum of the corresponding measures. To obtain a two-sided estimate, lower bounds on the coherence quantifiers are also of interest. Such bounds are naturally referred to as uncertainty relations for quantum coherence. We obtain new uncertainty relations for coherence quantifiers averaged with respect to a set of mutually unbiased bases (MUBs). To quantify the degree of coherence, the relative entropy of coherence and the geometric coherence are used. Further, we also derive novel state-independent uncertainty relations for a set of MUBs in terms of the min-entropy.