Absolutely entangled sets of pure states for bipartitions and multipartitions

A set of quantum states is said to be absolutely entangled when at least one state in the remains for any definition subsystems, i.e., choice global reference frame. In this work we investigate properties sets (AESs) pure states. For case a two-qubit system, present sufficient condition detect an AES, and use it construct families $N$ such that $N\ensuremath{-}3$ (the maximal possible number) remain subsystems. general bipartition $d={d}_{1}{d}_{2}$, prove $N>\ensuremath{\lfloor}({d}_{1}+1)({d}_{2}+1)/2\ensuremath{\rfloor}$ are AESs with Haar measure 1. Then, define multipartitions. We derive lower bound on number AES given multipartition, also explicit examples. particular, exhibit respect multipartitioning total system.