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arXiv:2607.04555 · Journal of Physics A Mathematical and Theoretical

Equiangular weighted frames and conical 2-designs with unequal traces

Katarzyna Siudzińska

quant-phSubmitted Jul 6, 2026• Announced Jul 7, 2026• 0 citations• Revised Jul 8, 2026 (v2)
v1 · Jul 6, 2026v2 · Jul 8, 2026
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Abstract

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.

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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.
Abstract Informationally overcomplete measurements find important applications in quantum tomography and quantum state estimation. The most popular are maximal sets of mutually unbiased bases, for which trace relations between measurement operators are well known. In this paper, we introduce a more general class of informationally overcomplete positive, operator-valued measure (POVMs) that are generated by equiangular tight frames of arbitrary rank. This class provides a generalization of equiangular measurements to non-projective POVMs, which include rescaled mutually unbiased measurements and bases. We provide a method of their construction, analyze their symmetry properties, and provide examples for highly symmetric cases. In particular, we find a wide class of generalized equiangular measurements that are conical two-designs, which allows us to derive the index of coincidence. Our results show benefits of considering a single informationally overcomplete measurement over informationally complete collections of POVMs.
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.
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.
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.
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.
Equioverlapping measurements, as a natural class of quantum measurements with the appealing property of equal overlap between any pair of measurement operators, generalize both the notions of equiangular tight frames (including von Neumann measurements) and symmetric informationally complete positive operator valued measures (SIC-POVMs). The structures of equioverlapping measurements in general dimensions are rather subtle and complicated. In this work, we reveal some structural properties of equioverlapping measurements which may be useful for constructing and classifying equioverlapping measurements. In particular, we obtain two bounds for the number of measurement operators in an equioverlapping measurement. We further illustrate how equioverlapping measurements go beyond SIC-POVMs in a nontrivial fashion with some illuminating examples in lower dimensions (two, three, and four). Finally, we present the challenging problem of fully classifying equioverlapping measurements and discuss some related perspectives.