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Hilbert–Schmidt quantum coherence in multi-qudit systems

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Abstract

Using Bloch’s parametrization for qudits (d-level quantum systems), we write the Hilbert–Schmidt distance (HSD) between two generic n-qudit states as an Euclidean distance between two vectors of observables mean values in \(\mathbb {R}^{\Pi _{s=1}^{n}d_{s}^{2}-1}\), where \(d_{s}\) is the dimension for qudit s. Then, applying the generalized Gell–Mann’s matrices to generate \(\mathrm{SU}(d_{s})\), we use that result to obtain the Hilbert–Schmidt quantum coherence (HSC) of n-qudit systems. As examples, we consider in detail one-qubit, one-qutrit, two-qubit, and two copies of one-qubit states. In this last case, the possibility for controlling local and non-local coherences by tuning local populations is studied, and the contrasting behaviors of HSC, \(l_{1}\)-norm coherence, and relative entropy of coherence in this regard are noticed. We also investigate the decoherent dynamics of these coherence functions under the action of qutrit dephasing and dissipation channels. At last, we analyze the non-monotonicity of HSD under tensor products and report the first instance of a consequence (for coherence quantification) of this kind of property of a quantum distance measure.

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Acknowledgements

This work was supported by the Brazilian funding agencies: Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq), processes 441875/2014-9 and 303496/2014-2, Coordenação de Desenvolvimento de Pessoal de Nível Superior (CAPES), process 6531/2014-08, and Instituto Nacional de Ciência e Tecnologia de Informação Quântica (INCT-IQ), process 2008/57856-6.

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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

    Jonas Maziero

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  1. Jonas Maziero
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Correspondence to Jonas Maziero.

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Maziero, J. Hilbert–Schmidt quantum coherence in multi-qudit systems. Quantum Inf Process 16, 274 (2017). https://doi.org/10.1007/s11128-017-1726-4

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