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An uncertainty view on complementarity and a complementarity view on uncertainty

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Abstract

Since the uncertainty about an observable of a system prepared in a quantum state is usually described by its variance, when the state is mixed, the variance is a hybrid of quantum and classical uncertainties. Besides that complementarity relations are saturated only for pure, single-quanton, quantum states. For mixed states, the wave–particle quantifiers never saturate the complementarity relation and can even reach zero for a maximally mixed state. So, to fully characterize a quanton it is not sufficient to consider its wave–particle aspect; one has also to regard its correlations with other systems. In this paper, we discuss the relation between quantum correlations and local classical uncertainty measures, as well as the relation between quantum coherence and quantum uncertainty quantifiers. We obtain a complete complementarity relation for quantum uncertainty, classical uncertainty, and predictability. The total quantum uncertainty of a d-paths interferometer is shown to be equivalent to the Wigner–Yanase coherence and the corresponding classical uncertainty is shown to be an entanglement monotone. The duality between complementarity and uncertainty is used to derive quantum correlations measures that complete the complementarity relations for \(l_{1}\)-norm and \(l_{2}\)-norm coherences. Besides, we show that Brukner–Zeilinger’s invariant information quantifies both the wave and particle characters of a quanton and we obtain a sum uncertainty relation for the generalized Gell-Mann’s matrices.

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Notes

  1. Entanglement monotones are nonnegative functions whose value does not increase under local operations and classical communication (LOCC) [32].

  2. Here, we will not define \(C_\mathrm{re}(\rho ) + S_\mathrm{vn}(\rho )\) as \({\mathcal {V}}(\rho )\) because the first is not a variance.

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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, 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, Rio Grande do Sul, 97105-900, Brazil

    Marcos L. W. Basso & Jonas Maziero

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  1. Marcos L. W. Basso
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Correspondence to Marcos L. W. Basso.

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Basso, M.L.W., Maziero, J. An uncertainty view on complementarity and a complementarity view on uncertainty. Quantum Inf Process 20, 201 (2021). https://doi.org/10.1007/s11128-021-03136-7

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