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Quantum Information Processing

Characterizing the average coherence via a broad class of informationally complete symmetric measurements

Liang Bao Tang, Fan Wu

Published Jan 18, 2023• 8 citations
Publisher
DOI
10.1007/s11128-022-03822-0

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A broad class of informationally complete symmetric measurements is introduced. It can be understood as a common generalization of symmetric, informationally complete positive operator-valued measures and mutually unbiased bases. Additionally, it provides a natural way to define two more families of mutually unbiased symmetric measurement operators in any finite dimension. We show a general method of their construction, together with an example of an optimal measurement. Finally, we analyze the properties of symmetric measurements and provide applications in entropic relations and entanglement detection.
Abstract The connection between uncertainty and entanglement is a prevalent topic in quantum information processing. Based on a broad class of informationally complete symmetric measurements, which can be viewed as a common generalization of symmetric, informationally complete positive operator-valued measures and mutually unbiased bases, a conical 2-design is calculated. This design plays a crucial role in quantum measurement theory. Subsequently, the relation between the uncertainty and the entanglement for a set of measurements is portrayed using conditional collision entropy. Furthermore, a tighter lower bound of the uncertainty relation is discussed according to the characterization of the entropic bound. Finally, the relation is applied to entanglement witnesses. It is demonstrated that the present results are unified and comprehensive.
Abstract Highly symmetric quantum measurements, such as mutually unbiased measurements (MUMs) and general symmetric informationally complete positive-operator-valued measures (GSIC-POVMs), play an important role in both foundational and practical aspects of quantum information theory. Recently, a broad class of symmetric measurements were introduced [K Siudzińska, (2022) Phys. Rev. A 105, 042209], which can be viewed as a common generalization of MUMs and GSIC-POVMs. In this work, the role of these symmetric measurements in entanglement detection is studied. More specifically, based on the correlation matrices defined via (informationally complete) symmetric measurements, a separability criterion for arbitrary dimensional bipartite systems is proposed. It is shown that the criterion is stronger than the method provided by Siudzińska, meanwhile, it can unify several popular separability criteria based on MUMs or GSIC-POVMs. Furthermore, using these (informationally complete) symmetric measurements, two efficient criteria are presented to detect the entanglement of tripartite quantum states. The detection power and advantages of these criteria are illustrated through several examples.