We introduce the task of locally shredding bipartite correlations, namely, the task of reducing total correlations in a given bipartite quantum state by means of local operations only, with the additional requirement that the amount of correlations that can be transferred to the environment during the decoupling process is upper bounded by a fraction (setting the ‘shredding condence level’) of those left in the output. Such a security constraint is needed in order to prevent the parties from simply preparing local states after having discharged their shares into the environment which could be afterwards accessed by an adversary. Here, in particular, we focus on the one-sided case, i. e. when only one party operates. We show that the task of one-sided correlation shredding divides the total amount of correlations into eliminable and ineliminable ones, that are those correlations that can (resp. cannot) be erased without violating the security constraint: this is done by proving tight upper and lower bounds on the amount of ineliminable correlations present in a given bipartite quantum state. Remarkably, such a division of total correlations turns out to be largely independent of the notion of entanglement, as there exist mixed entangled states possessing eliminable correlations only and separable states with ineliminable correlations. In the i.i.d. case, we show that ineliminable correlations are asymptotically measured by (the positive part of) coherent information, hence providing coherent information with an alternative interpretation. We nally speculate about the possibilities of ascribing the hereby proposed dichotomy eliminable/ineliminable correlations to some more fundamental classical/quantum dichotomy, and discuss various ways to extend the present analysis to more general scenarios.
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