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Quantitative wave–particle duality relations from the density matrix properties

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Abstract

We derive upper bounds for Hilbert–Schmidt’s quantum coherence of general states of a d-level quantum system, a qudit, in terms of its incoherent uncertainty, with the latter quantified using the linear and von Neumann’s entropies of the corresponding closest incoherent state. Similar bounds are obtained for Wigner–Yanase’s coherence. The reported inequalities are also given as coherence–populations trade-off relations. As an application example of these inequalities, we derive quantitative wave–particle duality relations for multi-slit interferometry. Our framework leads to the identification of predictability measures complementary to Hilbert–Schmidt’s, Wigner–Yanase’s, and \(l_{1}\)-norm quantum coherences. The quantifiers reported here for the wave and particle aspects of a quanton follow directly from the defining properties of the quantum density matrix (i.e., semi-positivity and unit trace), contrasting thus with most related results from the literature.

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Acknowledgements

This work was supported by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES), process 88882.427924/2019-01, by the Fundação de Amparo à Pesquisa do Estado do Rio Grande do Sul (FAPERGS), and by the Instituto Nacional de Ciência e Tecnologia de Informação Quântica (INCT-IQ), process 465469/2014-0.

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  1. Departamento de Física, Centro de Ciências Naturais e Exatas, Universidade Federal de Santa Maria, Avenida Roraima 1000, Santa Maria, RS, 97105-900, Brazil

    Marcos L. W. Basso, Diego S. S. Chrysosthemos & Jonas Maziero

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  1. Marcos L. W. Basso
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  2. Diego S. S. Chrysosthemos
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  3. Jonas Maziero
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Correspondence to Jonas Maziero.

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Basso, M.L.W., Chrysosthemos, D.S.S. & Maziero, J. Quantitative wave–particle duality relations from the density matrix properties. Quantum Inf Process 19, 254 (2020). https://doi.org/10.1007/s11128-020-02753-y

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