title: Should Entanglement Measures be Monogamous or Faithful?
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.
authors.affiliation: Física Teòrica: Informació i Fenòmens Quàntics, Universitat Autònoma de Barcelona, ES-08193 Bellaterra (Barcelona), Spain.|Institut Camille Jordan, Université Claude Bernard Lyon 1, 43 Boulevard du 11 Novembre 1918, 69622 Villeurbanne Cedex, France.
authors.forename: Cécilia
authors.initials: C
authors.lastname: Lancien
authors.affiliation: Física Teòrica: Informació i Fenòmens Quàntics, Universitat Autònoma de Barcelona, ES-08193 Bellaterra (Barcelona), Spain.
authors.forename: Sara
authors.initials: S
authors.lastname: Di Martino
authors.affiliation: Física Teòrica: Informació i Fenòmens Quàntics, Universitat Autònoma de Barcelona, ES-08193 Bellaterra (Barcelona), Spain.|Group of Applied Physics, University of Geneva, 1211 Geneva 4, Switzerland.|Institute for Quantum Optics and Quantum Information (IQOQI), Austrian Academy of Sciences, Boltzmanngasse 3, A-1090 Vienna, Austria.
authors.forename: Marcus
authors.initials: M
authors.lastname: Huber
authors.affiliation: SUPA and Department of Physics, University of Strathclyde, Glasgow G4 0NG, United Kingdom.
authors.forename: Marco
authors.initials: M
authors.lastname: Piani
authors.affiliation: School of Mathematical Sciences, The University of Nottingham, University Park, Nottingham NG7 2RD, United Kingdom.
authors.forename: Gerardo
authors.initials: G
authors.lastname: Adesso
authors.affiliation: Física Teòrica: Informació i Fenòmens Quàntics, Universitat Autònoma de Barcelona, ES-08193 Bellaterra (Barcelona), Spain.|ICREA-Institució Catalana de Recerca i Estudis Avançats, Passeig Lluis Companys 23, ES-08010 Barcelona, Spain.
authors.forename: Andreas
authors.initials: A
authors.lastname: Winter