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arXiv:2103.17145 · Journal of Physics Conference Series

Uncertainty and certainty relations for quantum coherence with respect to design-structured POVMs

Alexey E. Rastegin

Published Mar 1, 2021• 5 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1088/1742-6596/1847/1/012044

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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.
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.
Abstract The role of quantum coherence as an information resource has attracted increasing attention in recent years. Many protocols of quantum information science deal with specially selected states. Complete sets of mutually unbiased bases and symmetric informationally complete measurements are widely used in this regard. Other constructions were found to be useful, including projective designs and equiangular tight frames. As a rule, there are additional restrictions on the probabilities generated by quantum measurements assigned to such sets. Inequalities in terms of coherence quantifiers allow one to examine complementarity with respect to different sets. This study aims to examine uncertainty relations for coherence quantifiers averaged with respect to a set of mutually unbiased equiangular tight frames. To quantify the amount of coherence, quantum coherence quantifiers of the Tsallis type and the geometric coherence are used. The first case is induced by the Tsallis relative entropies. The derived inequalities are exemplified with equiangular tight frames of a ququart.
Abstract In quantum information science, systems with incomplete information are typically dealt with. To characterize quantumness from different viewpoints, several kinds of non-classical correlations should be described quantitatively. The concept of coherence within purely quantum framework is currently the subject of active research. A certain attention is paid to coherence quantifiers averaged with respect to a set of quantum ensembles or special measurements. Mutually unbiased bases and symmetric informationally complete measurements are important examples. We present uncertainty relations for quantum-coherence quantifiers based on the Tsallis relative 1/2-entropies. Together with mutually unbiased bases, the paper also deals with a measurement built of the states of an equiangular tight frame. The derived inequalities are exemplified with mutually unbiased bases and symmetric informationally complete measurement in two dimensions.
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.
Various measures have been suggested recently for quantifying the coherence of a quantum state with respect to a given basis. We first use two of these, the l1{l}_{1}l1​-norm and relative entropy measures, to investigate tradeoffs between the coherences of mutually unbiased bases. Results include relations between coherence, uncertainty, and purity; tight general bounds restricting the coherences of mutually unbiased bases; and an exact complementarity relation for qubit coherences. We further define the average coherence of a quantum state. For the l1{l}_{1}l1​-norm measure this is related to a natural ``coherence radius'' for the state and leads to a conjecture for an l2{l}_{2}l2​-norm measure of coherence. For relative entropy the average coherence is determined by the difference between the von Neumann entropy and the quantum subentropy of the state and leads to upper bounds for the latter quantity. Finally, we point out that the relative entropy of coherence is a special case of G-asymmetry, which immediately yields several operational interpretations in contexts as diverse as frame alignment, quantum communication, and metrology, and suggests generalizing the property of quantum coherence to arbitrary groups of physical transformations.