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

TL;DR

This paper introduces a family of entanglement monotones based on partial-norm measures that are not monogamous, challenging the assumption that all entanglement monotones must obey monogamy constraints.

Contribution

It defines new entanglement monotones using partial-norm measures that are concave but not strictly concave, demonstrating their non-monogamous nature and implications for axiomatic definitions.

Findings

Partial-norm of entanglement is not monogamous.

Strict concavity of reduced functions ensures monogamy.

Challenges existing axiomatic frameworks for entanglement measures.

Abstract

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.