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A Framework for Uncertainty Relations

Yunlong Xiao

Published Dec 9, 2016• 1 citation
OpenAlex

Abstract

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.

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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.
Learning physical properties of a quantum system is essential for the developments of quantum technologies. However, Heisenberg’s uncertainty principle constrains the potential knowledge one can simultaneously have about a system in quantum theory. Aside from its fundamental significance, the mathematical characterization of this restriction, known as ‘uncertainty relation’, plays important roles in a wide range of applications, stimulating the formation of tighter uncertainty relations. In this work, we investigate the fundamental limitations of variance-based uncertainty relations, and introduce several ‘near optimal’ bounds for incompatible observables. Our results consist of two morphologically distinct phases: lower bounds that illustrate the uncertainties about measurement outcomes, and the upper bound that indicates the potential knowledge we can gain. Combining them together leads to an uncertainty interval, which captures the essence of uncertainties in quantum theory. Finally, we have detailed how to formulate lower bounds for product-form variance-based uncertainty relations by employing entropic uncertainty relations, and hence built a link between different forms of uncertainty relations.
Uncertainty relations occupy a fundamental position in quantum mechanics. We propose stronger variance-based uncertainty relations for the product and sum of variances of two incompatible observables in a finite dimensional Hilbert space. It is shown that the new uncertainty relations provide near-optimal state-dependent bounds, which can be useful for quantum metrology, entanglement detection etc. in quantum information theory. It is further shown that the uncertainty relations are related to the spreads of the distribution of measurement outcomes caused by incompatible observables. Intuitively, this means that the ability of learning the distribution has both the upper and lower bounds. Combination of these bounds provides naturally an uncertainty interval which captures the essence of uncertainty in quantum theory. Finally, we explain how to employ entropic uncertainty relations to derive lower bounds for the product of variances of incompatible observables.
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.
The uncertainty principle bounds the uncertainties about incompatible measurements, clearly setting quantum theory apart from the classical world. Its mathematical formulation via uncertainty relations, plays an irreplaceable role in quantum technologies. However, neither the uncertainty principle nor uncertainty relations can fully describe the complementarity between quantum measurements. As an attempt to advance the efforts of complementarity in quantum theories, we formally propose a complementary information principle, significantly extending the one introduced by Heisenberg. First, we build a framework of black box testing consisting of pre- and post-testing with two incompatible measurements, introducing a rigorous mathematical expression of complementarity with definite information causality. Second, we provide majorization lower and upper bounds for the complementary information by utilizing the tool of semidefinite programming. In particular, we prove that our bounds are optimal under majorization due to the completeness of the majorization lattice. Finally, as applications to our framework, we present a general method to outer-approximating all uncertainty regions and also establish fundamental limits for all qualified joint uncertainties.
The uncertainty principle can be expressed in entropic terms, also taking into account the role of entanglement in reducing uncertainty. The information exclusion principle bounds instead the correlations that can exist between the outcomes of incompatible measurements on one physical system, and a second reference system. We provide a more stringent formulation of both the uncertainty principle and the information exclusion principle, with direct applications for, e.g., the security analysis of quantum key distribution, entanglement estimation, and quantum communication. We also highlight a fundamental distinction between the complementarity of observables in terms of uncertainty and in terms of information.
In this paper we revisit the quantitative formulation of the Heisenberg uncertainty principle. The primary version of this principle establishes the impossibility of refined simultaneous measurement of position x and momentum u for a (1-dimensional) quantum particle in terms of variances: <||x||^2> <||u||^2> ≥ 1/4. Since this inequality applies provided each variance exists, some authors proposed entropic versions of this principle as an alternative (employing Shannon's or Rényi's entropies). As another alternative, we consider moments-based formulations and show that inequalities involving moments of orders other than 2 can be found. Our procedure is based on the Rényi entropic versions of the Heisenberg relation together with the search for the maximal entropy un- der statistical moments' constraints ( <||x||^a> and <||u||^b> ). Our result improves a relation proposed very recently by Dehesa et al. [1] where the same approach was used but starting with the Shannon version of the entropic uncertainty relation. Furthermore, we show that when a = b, the best bound we can find with our approach coincides with that of Ref. [1] and, in addition, for a = b = 2 the variance-based Heisenberg relation is recovered. Finally, we illustrate our results in the cases of d-dimensional hydrogenic systems.