Non-commutative measure of quantum correlations under local operations

Abstract

We study some desirable properties of recently introduced measures of quantum correlations based on the amount of non-commutativity quantified by the Hilbert–Schmidt norm (Guo in Sci Rep 6:25241, 2016; Majtey et al. in Quantum Inf Process 16:226, 2017). Specifically, we show that: (1) for any bipartite (\(A+B\)) state, the measures of quantum correlations with respect to subsystem A are non-increasing under any local commutative preserving operation on subsystem A, and (2) for Bell-diagonal states, the measures are non-increasing under arbitrary local operations on B. Our results accentuate the potentialities of such measures and exhibit them as valid monotones in a resource theory of quantum correlations with free operations restricted to the appropriate local channels.

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Notes

  1. 1.

    A LCPO corresponds to a map \(\Delta [\cdot ]\) that is completely positive trace preserving and preserves the commutativity [36], that is, \([\Delta [\rho ],\Delta [\sigma ]]=0 \ \ \forall \;\rho ,\sigma \; \text {such that} \; [\rho ,\sigma ]=0.\)

  2. 2.

    Recall that a map \(\Phi [\cdot ]\) is said to be unital if \(\Phi [{\mathbb {I}}]={\mathbb {I}}\), whereas a completely decohering map \(\Phi [\cdot ]\) is such that \(\Phi [\rho ]=\sum _ip_i \left| {i}\right\rangle \left\langle {i}\right| \), for some orthonormal basis \(\{\left| {i}\right\rangle \}\) and (state-dependent) probabilities \(\{p_i\}\).

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Acknowledgements

D. B. and A. P. M. acknowledge the Argentinian agency SeCyT-UNC and CONICET for financial support. D. B. has a fellowship from CONICET. A. V. H. gratefully acknowledges financial support from DGAPA, UNAM through project PAPIIT IA101918.

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Correspondence to A. Valdés-Hernández.

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Bussandri, D.G., Majtey, A.P. & Valdés-Hernández, A. Non-commutative measure of quantum correlations under local operations. Quantum Inf Process 18, 47 (2019). https://doi.org/10.1007/s11128-018-2154-9

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Keywords

  • Quantum correlation measures
  • Non-commutativity quantum discord
  • Local operations
  • Local commutative preserving operations