Abstract
The notions of k-separability and k-producibility are useful and expressive tools for the characterization of entanglement in multipartite quantum systems, when a more detailed analysis would be infeasible or simply needless. In this work we reveal a partial duality between them, which is valid also for their correlation counterparts. This duality can be seen from a much wider perspective, when we consider the entanglement and correlation properties which are invariant under the permutations of the subsystems. These properties are labeled by Young diagrams, which we endow with a refinement-like partial order, to build up their classification scheme. This general treatment reveals a new property, which we call k-stretchability, being sensitive in a balanced way to both the maximal size of correlated (or entangled) subsystems and the minimal number of subsystems uncorrelated with (or separable from) one another.
Highlights
In this work we investigated the partial correlation and entanglement properties which are invariant under the permutations of the subsystems
We show the ξ-LO monotonicity of the ξcorrelation (5a), from which the LO monotonicity of the ξ-correlation (10a) follows, because the latter one is a minimum of some of the former ones
Note that the monotonicity shown here is a particular case of a much more general property of monotone distance based geometric measures [65] in resource theories [75]: the monotonicity with respect to free maps, by which the set of free states is mapped onto itself
Summary
We recall the structure of the classification and quantification of multipartite correlation and entanglement [14, 40, 41]. Our goal is to do this in the way sufficient to see how the permutation invariant properties can be formulated parallel to this
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