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Kirkwood–Dirac quasiprobabilities for measurements related to mutually unbiased equiangular tight frames

Alexey E. Rastegin

Published Apr 14, 2025• 5 citations
Publisher

Abstract

Protocols of quantum information science often realize in terms of specially selected states. In particular, such states are used to perform measurements at the final stage of a protocol. This study aims to explore measurements assigned to a mutually unbiased-equiangular tight frame. The utilized method deals with Kirkwood–Dirac quasiprobabilities, which are increasingly used in contemporary research. These quasiprobabilities constitute a matrix that can be linked to unravelings of certain quantum channels. Using states of the given frame to build principal Kraus operators leads to quasiprobabilities that represent the measured state. The structure of a mutually unbiased-equiangular tight frame allows one to characterize entropies associated with a particular unraveling. To do this, we estimate some of the Schatten and Ky Fan norms of the matrix consisted of quasiprobabilities. New uncertainty relations in terms of Rényi and Tsallis entropies follow from the obtained inequalities. A utility of the presented inequalities is exemplified with mutually unbiased bases of a qubit and equiangular tight frames of a ququart.

DOI
10.1063/5.0254873

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Protocols of quantum information science often realize in terms of specially selected states. In particular, such states are used to perform measurements at the final stage of a protocol. This study aims to explore measurements assigned to a mutually unbiased-equiangular tight frame. The utilized method deals with Kirkwood–Dirac quasiprobabilities, which are increasingly used in contemporary research. These quasiprobabilities constitute a matrix that can be linked to unravelings of certain quantum channels. Using states of the given frame to build principal Kraus operators leads to quasiprobabilities that represent the measured state. The structure of a mutually unbiased-equiangular tight frame allows one to characterize entropies associated with a particular unraveling. To do this, we estimate some of the Schatten and Ky Fan norms of the matrix consisted of quasiprobabilities. New uncertainty relations in terms of Rényi and Tsallis entropies follow from the obtained inequalities. A utility of the presented inequalities is exemplified with mutually unbiased bases of a qubit and equiangular tight frames of a ququart.
An issue which has attracted increasing attention in contemporary researches are Kirkwood--Dirac quasiprobabilities. List of their use includes many questions of quantum physics. Applications of complex tight frames in quantum information science were recently demonstrated. It is shown in this paper that quasiprobabilities naturally appear in the context of unravelings of a quantum channel. Using vectors of the given tight frame to build principal Kraus operators generates quasiprobabilities with interesting properties. For an equiangular tight frame, we characterize the Hilbert--Schmidt and spectral norms of the matrix consisted of quasiprobabilities. Hence, novel uncertainty relations in terms of Rényi and Tsallis entropies are obtained. New inequalities for characterizing the location of eigenvalues are derived. They give an alternative to estimating on the base of Geršgorin's theorem. A utility of the presented inequalities is exemplified with symmetric informationally complete measurement in dimension two.
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 theory, one needs to consider systems with incomplete information. To estimate a quantum system as an information resource, one uses various characteristics of non-classical correlations. Currently, much attention is paid to coherence quantifiers averaged over a set of specially selected states. In particular, mutually unbiased bases, symmetric informationally complete measurements, and some of their generalizations are of importance in this regard. The aim of the present study is to derive uncertainty relations for coherence quantifiers based on divergences of the Tsallis type. The obtained inequalities concern quantifiers averaged over a set of mutually unbiased bases and a set of states that form an equiangular tight frame.
Building on recent significant interest in entropic uncertainty relations for quantum measurements, this work explores the entropic uncertainty relation for the mutually unbiased equiangular tight frame assigned to quantum ttt-designs. Combining the mathematical structures of quantum measurements with those of quantum designs, uncertainty relations are derived in terms of Tsallis and R\'enyi $\ensuremath{\alpha}$ entropies, and rigorous theoretical analysis demonstrates the tightness of the derived entropic lower bounds. Furthermore, a specific relation between quantum design and the index of coincidence is inferred from the information diagrams, which is crucial for deriving the novel entropic uncertainty relation. The results reveal that the entropic lower bounds can be optimized by adjusting the parameters of the quantum ttt-design, broadening the theoretical framework of the entropic uncertainty relation and providing a practical tool for quantum information tasks.
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.
Abstract The detection of quantum entanglement represents a fundamental challenge in quantum measurement theory within the context of quantum information science. In this work, we propose a family of separability criteria based on mutually unbiased equiangular tight frames [IEEE Trans. Inf. Theory 67 , 1656–1667 (2021)], which constitute a natural generalization of both mutually unbiased bases and equiangular tight frames. The proposed criteria are applicable to a broad class of quantum systems, including bipartite systems with equal or unequal subsystem dimensions, as well as multipartite systems with heterogeneous dimensional components. In the examined cases, namely isotropic, Bell-diagonal, and Werner states, the proposed entanglement detection criteria demonstrate superior performance relative to several established methods. Crucially, these criteria rely solely on a small set of local measurements, which significantly enhances their experimental feasibility for the characterization of entanglement in unknown quantum states. Overall, the MUETF-based criteria provide more robust and versatile tools for entanglement verification across diverse quantum architectures.