Why the Entanglement of Formation is not generally monogamic

TL;DR

This paper investigates why the Entanglement of Formation (EOF) generally does not satisfy monogamy of entanglement, revealing conditions under which EOF can be monogamous and relating it to classical-quantum correlation discrepancies.

Contribution

The paper identifies conditions for EOF to be monogamous and links its monogamy properties to the balance between quantum and classical correlations.

Findings

EOF can be monogamous under specific conditions.

Negative _{ABC} indicates quantum correlation dominance.

EOF's relation to Squashed Entanglement clarifies its monogamy properties.

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.