"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.Introduction. Entanglement is a quintessential manifestation of quantum mechanics [1,2]. The study of entanglement and its distribution reveals fundamental insights into the nature of quantum correlations [3], on the properties of manybody systems [4,5], and on possibilities and limitations for quantum-enhanced technologies [6]. A particularly interesting feature of entanglement is known as monogamy [7], that is, the impossibility of sharing entanglement unconditionally across many subsystems of a composite quantum system.In the clearest manifestation of monogamy, if two parties A and B with the same (finite) Hilbert space dimension are maximally entangled, then their state is a pure state |Φ AB [8], and neither of them can share any correlation-let alone entanglement-with a third party C, as the only physically allowed pure states of the tripartite system ABC are product states |Φ AB ⊗ |Ψ C . Consider now the more realistic case of A and B being in a mixed, partially entangled state ρ AB . It is then conceivable for more parties to get a share of such entanglement. Namely, a state ρ AB on a Hilbert space H A ⊗ H B is termed "n-shareable" with respect to subsystem B if it admits a symmetric n-extension, i.e. a state ρ AB 1 ...B n on H A ⊗ H ⊗n B invariant under permutations of the subsystems B 1 , . . . , B n and such that the marginal state of A and any B j amounts to ρ AB . While even an entangled state can be shareable up to some number of extensions, a seminal result is that a state ρ AB is n-shareable for all n 2 if and only if it is separable, that is, no entangled state can be infinitely-shareable [7,[9][10][11][12]. This statement formalizes exactly the monogamy of entanglement (in an asymptotic setting), and has many important implications, including the equivalence between asymptotic quantum cloning and state e...

We analyze the distinguishability norm on the states of a multi-partite system, defined by local measurements. Concretely, we show that the norm associated to a tensor product of sufficiently symmetric measurements is essentially equivalent to a multi-partite generalisation of the non-commutative 2 -norm (aka Hilbert-Schmidt norm): in comparing the two, the constants of domination depend only on the number of parties but not on the Hilbert spaces dimensions.We discuss implications of this result on the corresponding norms for the class of all measurements implementable by local operations and classical communication (LOCC), and in particular on the leading order optimality of multi-party data hiding schemes. I. DISTINGUISHABILITY NORMSThe task of distinguishing quantum states from accessible experimental data is at the heart of quantum information theory, appearing right at its historical beginnings -see [10, 11], and [15] for general reference. Indeed, the special case on which we are focussing in this paper, the discrimination of two states, is the generalisation of hypothesis testing in classical statistics. There, the optimal discrimination between two hypotheses, modelled as (for simplicity: discrete) probability distributions P 0 and P 1 , with prior probabilities q and 1 − q, respectively, is given by the maximum likelihood rule [7]. The minimum error probability is thus given byIn this paper, we shall denote by the same symbol its non-commutative generalisation ∆ 1 = Tr |∆|, i.e. the sum of the singular values of ∆, also known as trace norm.Owing to the particular role played by measurement in quantum mechanics, however, any restriction on the set of available measurements leads to a specific norm on density operators: any decision in the discrimination task must be based on measurement results. Specifically, let the two hypotheses be two quantum states (density operators) ρ 0 and ρ 1 on some Hilbert space H, with prior probabilities q and 1 − q, respectively. A generic measurement M , i.e. a positive operator valued measure (POVM, aka partition of unity), is given by positive semidefinite operators M x ≥ 0, s.t.x∈X M x = 1 1.