Non-commutative measure of quantum correlations under local operations

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 ($$A+B$$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.