k-stretchability of entanglement, and the duality of k-separability and k-producibility

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 (or separable) from one another.

Entanglement of Local Hidden States

“…. , |B |Λ| |} and height h = |Λ| then identify their w-producibility and h-separability [12]. Combining these quantities, we can introduce classes of (w, h)-separable states whose separable partitions have a width that does not exceed w and a height no smaller than h [43].…”

“…The classification of quantum correlations is crucial for our understanding of the resources that enable quantum information tasks [1,2,3,4,5,6,7]. Two of the main challenges in this field are the characterization of entanglement in multipartite systems [8,9,10,11,12] and the identification of nonclassical correlations stronger than entanglement that allow to relax the assumptions to be made about the system of interest [3,4,5,7]. Bell nonlocality, the strongest form of quantum correlations, allow for fully device-independent entanglement detection [13,4].…”

Experimentally accessible non-separability criteria for multipartite entanglement structure detection

“…Recently, the structure of partial separability and multipartite entanglement has been investigated with the goal of introducing the mathematical formalism to best identify and describe its rich hierarchy [8][9][10][11]. For bipartite systems, the fact that there is only one possible partition makes its definition relatively simple.…”

“…Following Refs. [9][10][11], we may call this notion of entanglement, namely with respect to specific set partitions, level-I multipartite entanglement.…”

Partial separability/entanglement violates distributive rules

“…More recently, the convex hulls of two of them have also been considered together with intersections and complements of convex sets arising in the way [12][13][14], leading to the description of the hierarchy of the intersections [15]. See also [16][17][18] for further developments. Recall that the intersection and convex hull of the convex sets of three basic biseparable states give rise to fully biseparable and biseparable states, respectively.…”