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