Enlarging the concept of additivity of quantum resources

Abstract

For future quantum computation developments, it is important to address the issue of quantifying the amount of resources of a certain number of copies of quantum states. This is a hard problem, often involving optimizations over Hilbert spaces of large dimensions. We propose a way to circumvent the direct evaluation of such quantities, provided that the employed quantifiers satisfy a self-similarity property, which we call scalability. This property is in essence a constraint on the way certain quantities can scale with the number of copies of a given system. If analyticity is assumed, recursive relations can be derived for the Maclaurin series of 𝒠(ρ ⊗N ), which enables us to determine its possible functional forms. Our approach sets possible ways in which a broad class of quantum functions can scale with the number of copies of a quantum state, describing very simply the non-additivity of some relevant resources, such as quantum coherence and distillable entanglement. A generalization of linear scalable functions is formalized.