ar X iv : q ua nt - p h / 07 01 12 4 v 1 1 7 Ja n 20 07 Entanglement and Subsystems , Entanglement beyond Subsystems , and All That ∗

Entanglement plays a pervasive role nowadays throughout quantum information science, and at the same time provides a bridging notion between quantum information science and fields as diverse as condensed-matter theory, quantum gravity, and quantum foundations. In recent years, a notion of Generalized Entanglement (GE) has emerged [H. Barnum et al, Phys. Rev. A 68, 032308 (2003); L. Viola et al, Contemp. Math. 381, 117 (2005)], based on the idea that entanglement may be directly defined through expectation values of preferred observables – without reference to a preferred subsystem decomposition. Preferred observables capture the physically relevant point of view, as defined by dynamical, operational, or fundamental constraints. While reducing to the standard entanglement notion when preferred observables are restricted to arbitrary local observables acting on individual subsystems, GE substantially expands subsystem-based entanglement theories, in terms of both conceptual foundations and range of applicability. Remarkably, the GE framework allows for non-trivial entanglement to exist within a single, indecomposable quantum system, demands in general a distinction between quantum separability and absence of entanglement, and naturally extends to situations where existing approaches may not be directly useful – such as entanglement in arbitrary convex-cones settings and entanglement for indistinguishable quantum particles. In this paper, we revisit the main motivations leading to GE, and summarize the accomplishments and prospects of the GE program to date, with an eye toward conceptual developments and implications. In particular, we explain how the GE approach both shares strong points of contact with abstract operational quantum theories and, ultimately, calls for an observer-dependent redefinition of concepts like locality, completeness, and reality in quantum theory.