Should Entanglement Measures be Monogamous or Faithful?

TL;DR

This paper investigates the conditions under which entanglement measures obey quantitative monogamy relations, revealing that many measures fail to do so universally, but can satisfy dimension-dependent monogamy constraints.

Contribution

It formalizes the criteria for monogamy of entanglement measures and demonstrates the limitations of additive measures in satisfying universal monogamy relations.

Findings

Many entanglement measures are not monogamous in high dimensions.

Additive and normalized measures cannot satisfy universal monogamy relations.

Dimension-dependent monogamy relations can restore monogamy for certain measures.

Abstract

"Is entanglement monogamous?" asks the title of a popular article [B. Terhal, IBM J. Res. Dev. 48, 71 (2004)], celebrating C. H. Bennett's legacy on quantum information theory. While the answer is affirmative in the qualitative sense, the situation is less clear if monogamy is intended as a quantitative limitation on the distribution of bipartite entanglement in a multipartite system, given some particular measure of entanglement. Here, we formalize what it takes for a bipartite measure of entanglement to obey a general quantitative monogamy relation on all quantum states. We then prove that an important class of entanglement measures fail to be monogamous in this general sense of the term, with monogamy violations becoming generic with increasing dimension. In particular, we show that every additive and suitably normalized entanglement measure cannot satisfy any nontrivial general monogamy relation while at the same time faithfully capturing the geometric entanglement structure of the fully antisymmetric state in arbitrary dimension. Nevertheless, monogamy of such entanglement measures can be recovered if one allows for dimension-dependent relations, as we show explicitly with relevant examples.