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International Journal of Theoretical Physics

Uncertainty Relation of Quantum Coherence Measure Based on Hellinger Distance

Xueqiang Li, Fu-Gang Zhang

Published Jan 9, 2024• 3 citations
Publisher
DOI
10.1007/s10773-023-05537-z

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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).

Physica Scripta•Mar 27, 2026•0 citations
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).