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Proceedings of the Royal Society A Mathematical Physical and Engineering Sciences

Some properties of measurements assigned to conical two-designs

Alexey E. Rastegin

Published Jul 15, 2026• 0 citations
Publisher

Abstract

Abstract Quantum measurements are an important step in implementing protocols of quantum information processing. Conical two-designs were raised as a tool for these purposes. The current study considers some valuable features of measurements assigned to conical two-designs. Kirkwood–Dirac quasiprobabilities often give a useful alternative to other quasiprobability distributions. There are especially interesting cases of conical two-designs. Two of them allow us to evaluate the Hilbert–Schmidt norm of the corresponding matrix. So, we take a conical two-design with operators of the same trace, and a conical two-design from generalized equiangular measurements. Any conical two-design can induce separability criteria. This consideration reveals a few aspects not addressed in the literature. Uncertainty relations follow for the case of operators with the same trace. This research completes previous studies of uncertainty relations from conical two-designs.

DOI
10.1098/rspa.2025.0827

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Abstract Quantum measurements are an important step in implementing protocols of quantum information processing. Conical two-designs were raised as a tool for these purposes. The current study considers some valuable features of measurements assigned to conical two-designs. Kirkwood–Dirac quasiprobabilities often give a useful alternative to other quasiprobability distributions. There are especially interesting cases of conical two-designs. Two of them allow us to evaluate the Hilbert–Schmidt norm of the corresponding matrix. So, we take a conical two-design with operators of the same trace, and a conical two-design from generalized equiangular measurements. Any conical two-design can induce separability criteria. This consideration reveals a few aspects not addressed in the literature. Uncertainty relations follow for the case of operators with the same trace. This research completes previous studies of uncertainty relations from conical two-designs.
Abstract Quantum measurements are important tools in quantum information, represented by positive, operator-valued measures. A wide class of symmetric measurements is given via generalized equiangular measurements (GEAMs) that form conical 2-designs. We show that only two positive constants are needed to fully characterize a variety of important quantum measures constructed from such operators. Examples are given for entropic uncertainty relations, the Brukner–Zeilinger invariants, quantum coherence, quantum concurrence, and the Schmidt-number criterion for entanglement detection. Our results indicate that similar relations may also hold for conical 2-designs from beyond the class of GEAMs.
Conical 2-designs are families of positive operators possessing crucial symmetry properties that allow them to efficiently characterize important quantum measures. However, little is known about their general structures and properties. We propose a construction method of conical 2-designs whose elements are not of equal traces, based on a generalization of equiangular weighted frames to non-projective operators. We provide a detailed analysis of their properties, relations to quantum measurements, and applications, together with examples that go beyond the known classes. This constitutes a major step toward a full characterization of informationally overcomplete quantum measurements.
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.
Quantum coherence is an important quantum resource which plays a pivotal role in the field of quantum information. Based on metric adjusted skew information, we define a measure of quantum uncertainty to study average coherence under conical 2-designs generalized equiangular measurements, and prove the equivalence of this measure to the scaled average coherence based on metric adjusted skew information under a set of unitary groups, operator orthonormal bases, and mutually unbiased bases. We also derive two trade-off relations by this measure and solve a conjecture. Furthermore, we give two entanglement criteria by this measure and conical 2-designs generalized equiangular measurement, respectively, and illustrate the effectiveness of them by explicit examples.
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.
Quantum measurements play a fundamental role in quantum information. Therefore, increasing efforts are being made to construct symmetric measurement operators for qudit systems. A wide class of projective measurements corresponds to complex projective 2-designs, which include symmetric, informationally complete (SIC) POVMs and mutually unbiased bases (MUBs). In this paper, we establish a one-to-one correspondence between conical 2-designs and mutually unbiased generalized equiangular tight frames, both of which are common generalizations of SIC POVMs and MUBs to operators of arbitrary rank. It turns out that there exist rich families of operators that belong to only one of those two classes. This raises important questions about which symmetries have to be preserved for applicational prominence.