Why entanglement of formation is not generally monogamous

Abstract

Unlike 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 with a third system $C$. This concept, known as the monogamy of entanglement, is manifest when the entanglement of $A$ with a pair $BC$ can be divided as contributions of the entanglement between $A$ and $B$ and $A$ and $C$, plus a term ${\ensuremath{\tau}}_{ABC}$ 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 the EOF to become a monogamous measure, i.e., for ${\ensuremath{\tau}}_{ABC}\ensuremath{\ge}0$. The relation derived is connected to the discrepancy between quantum and classical correlations, ${\ensuremath{\tau}}_{ABC}$ being 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 the EOF to the squashed sntanglement, an entanglement measure that is always monogamous.