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arXiv:2112.12375 · Proceedings of the Royal Society A Mathematical Physical and Engineering Sciences

Entropic uncertainty relations from equiangular tight frames and their applications

Alexey E. Rastegin

Published Jun 1, 2023• 26 citations
arXivHTMLPDF

Abstract

Finite tight frames are interesting in various topics including questions of quantum information. Each complex tight frame leads to a resolution of the identity in the Hilbert space. Symmetric informationally complete measurements are a special class of equiangular tight frames. Applications of such frames in quantum physics deserve more attention than they have obtained. We derive uncertainty relations for a quantum measurement assigned to an equiangular tight frame. Main results follow from estimation of the corresponding index of coincidence. State-dependent and state-independent formulations are both addressed. Also, we discuss applications of considered measurements to detect entanglement and other correlations.

DOI
10.1098/rspa.2022.0546

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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.
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Abstract Improvement of the detection ability of quantum entanglement is one of the essential tasks in quantum computing and quantum information. Finite tight frames play a fundamental role in a wide variety of areas and, generally, each application requires a specific class of frames and is closely related to quantum measurement. It is worth noting that a maximal set of complex equiangular vectors is closely related to a symmetric informationally complete measurement. Hence, our goal in this work is to propose a series of separability criteria assigned to a finite tight frame and some well-known inequalities in different quantum systems, respectively. In addition, some tighter criteria to detect entanglement for many-body quantum states are presented in arbitrary dimensions. Finally, the effectiveness of the proposed entanglement detection criteria is illustrated through some detailed examples.
In quantum information theory, measurements are represented by positive, operator-valued measures. In this paper, we use the operators corresponding to generalized equiangular measurements to construct positive maps. Their positivity follows from the properties of index of coincidence for few equiangular tight frames. These maps give rise to entanglement witnesses, which include as special cases many important classes considered in the literature. Additionally, we introduce separability criteria based on the correlation matrix and analyze them for various types of measurements.
Abstract Finite tight frames play an important role in miscellaneous areas, including quantum information theory. Here we apply a class of tight frames, equiangular tight frames, to address the problem of detecting the entanglement of bipartite states. Here we derive some entanglement criteria based on positive operator-valued measurements built from equiangular tight frames. We also present a class of entanglement witnesses based on the equiangular tight frames. At last, we generalize the entanglement criterion for bipartite systems to multipartite systems.