Partial-Norm of Entanglement: Entanglement Monotones That are not Monogamous

Quantum entanglement is known to be monogamous, i.e., it obeys strong constraints on how the entanglement can be distributed among multipartite systems. Almost all the entanglement monotones so far are shown to be monogamous. We explore here a family of entanglement monotones with the reduced functions are concave but not strictly concave and show that they are not monogamous. They are defined by four kinds of the ``partial-norm'' of the reduced state, which we call them \textit{partial-norm of entanglement}, minimal partial-norm of entanglement, reinforced minimal partial-norm of entanglement, and \textit{partial negativity}, respectively. This indicates that, the previous axiomatic definition of the entanglement monotone needs supplemental agreement that the reduced function should be strictly concave since such a strict concavity can make sure that the corresponding convex-roof extended entanglement monotone is monogamous. Here, the reduced function of an entanglement monotone refers to the corresponding function on the reduced state for the measure on bipartite pure states.