Why the Entanglement of Formation is not generally monogamic

Abstract

Differently from correlation of classical systems, entanglement of quantum systems cannot be distributed at will - if one system A is maximally entangled with another system B, it cannot be entangled at all to a third system C. This concept, known as the monogamy of entanglement, manifests when the entanglement of A with a pair BC, can be divided as contributions of entanglement between A and B and A and C, plus a term \tau_{ABC} involving genuine tripartite entanglement and so expected to be always positive. A very important measure in Quantum Information Theory, the Entanglement of Formation (EOF), fails to satisfy this last requirement. Here we present the reasons for that and show a set of conditions that an arbitrary pure tripartite state must satisfy for EOF to become a monogamous measure, ie, for \tau_{ABC} \ge 0. The relation derived is connected to the discrepancy between quantum and classical correlations, being \tau_{ABC} negative whenever the quantum correlation prevails over the classical one. This result is employed to elucidate features of the distribution of entanglement during a dynamical evolution. It also helps to relate all monogamous instances of EOF to the Squashed Entanglement, an always monogamous entanglement measure.