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Quantum Information Processing

The relation between entanglement measure and coherence measure based on Hellinger distance

Yaxue Liu, Lili Yang, Donghua Yan

Published Mar 31, 2022• 3 citations
Publisher
DOI
10.1007/s11128-022-03465-1

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Quantum coherence and nonclassical correlation are key features of a quantum world. Quantifying coherence and nonclassical correlation are two key tasks in quantum information theory. First, we present a bona fide measure of quantum coherence by utilizing the Hellinger distance. This coherence measure is proven to fulfill all the criteria of a well-defined coherence measure, including the strong monotonicity in the resource theories of quantum coherence. In terms of this coherence measure, the distribution of quantum coherence in multipartite systems is studied and a corresponding polygamy relation is proposed. Its operational meanings and the relations between the generation of quantum correlations and the coherence are also investigated. Moreover, we present the Hellinger distance-based measure of nonclassical correlation, which not only inherits the nice properties of the Hellinger distance including contractivity, but also shows a powerful analytic computability for a large class of quantum states. We show that there is an explicit tradeoff relation satisfied by the quantum coherence and this nonclassical correlation.
Abstract Quantum coherence, a fundamental quantum resource, is critical for quantum computing, communication, and metrology. Hellinger-distance-based coherence enables robust quantum superposition quantification; however its basis-dependent behavior hinders practical use in quantum information processing. To address this, this study explores the trade-off relations of Hellinger-distance-based coherence in qubits, with core contributions: calculating its upper bound (max achievable value for any qubit state); deriving its complementary relations with geometric mixedness (higher mixedness reduces coherence); and establishing its trade-offs under mutually unbiased bases. For applications, we link the minimum error probability of pure state discrimination (via least square measurement) to geometric purity (optimizing discrimination), and connect Hellinger-distance-based coherence to nonlocal advantage (critical for quantum steering).
By formally generalizing the classical Hellinger distance and affinity on the space of probability densities, we introduce their quantum analog on the quantum state space consisting of all density operators. We show that the infinitesimal form of the quantum Hellinger distance is the skew information introduced by Wigner and Yanase in 1963. We compare the Hellinger distance and affinity with the Bures distance and fidelity, and establish a variety of their fundamental properties. We further apply them to characterizing entanglement and to establishing some unusual uncertainty relations relating nonsimultaneous measurements of a single observable in two different quantum states.