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Predictability as a quantum resource

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Abstract

Just recently, complementarity relations (CRs) have been derived from the basic rules of Quantum Mechanics. The complete CRs are equalities involving quantum coherence, C, quantum entanglement, and predictability, P. While the first two are already quantified in the resource theory framework, such a characterization lacks for the last. In this article, we start showing that, for a system prepared in a state \(\rho \), P of \(\rho \), with reference to an observable X, is equal to C, with reference to observables mutually unbiased (MU) to X, of the state \(\Phi _{X}(\rho )\), which is obtained from a non-revealing von Neumann measurement (NRvNM) of X. We also show that \(P^X(\rho )>C^{Y}(\Phi _{X}(\rho ))\) for observables X, Y not MU. Afterwards, we provide quantum circuits for implementing NRvNMs and use these circuits to experimentally test these (in)equalities using the IBM’s quantum computers. Furthermore, we give a resource theory for predictability, identifying its free quantum states and free quantum operations and discussing some predictability monotones. Besides, after applying one of these predictability monotones to study bipartite systems, we discuss the relation among the resource theories of quantum coherence, predictability, and purity.

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Acknowledgements

This work was supported by Universidade Federal do ABC (UFABC), process 23006.000123/2018-23, and by the Instituto Nacional de Ciência e Tecnologia de Informação Quântica (INCT-IQ), process 465469/2014-0. We thank Renato Moreira Angelo for the question we answered in this article and for illuminating conversations. We also thank Alexandre Camacho Orthey Junior for discussions regarding the implementation of local non-revealing von Neumann measurements through global unitaries.

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Authors and Affiliations

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

    Marcos L. W. Basso & Jonas Maziero

  2. Centro de Ciências Naturais e Humanas, Universidade Federal do ABC, Avenida dos Estados 5001, 09210-580, Santo André, São Paulo, Brazil

    Marcos L. W. Basso

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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. Predictability as a quantum resource. Quantum Inf Process 21, 187 (2022). https://doi.org/10.1007/s11128-022-03503-y

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