Distributions of asymptotic transformation rates among quantum states

In quantum resource theories, manipulating and transformation resource states are often challenging due to the presence of noise. Resource manipulation procedures, from high resource states ρ to low resource states ρ', involving asymptotically many copies of states can be considered to overcome the problem. Here, the asymptomatic transformation rate R(ρ→ρ') can characterize the corresponding quantum manipulation power, and can be calculated as the ratio of the copy number of initial states to the copy number of target states. Generally, exact computations of asymptotic transformation rates are challenging, so it is important to establish rigorous and computable bounds on them. Recently, Ganardi et al. show that the transformation rate to any pure state is superadditive for the distillable entanglement. However, it has remained a question whether the transformation rate to any noise state is also superadditive in the general resource theory. Firstly, we study the general superadditive inequalities satisfied by the transformation rate R(ρ→ρ') to any noise stateρ'. In any multiple quantum resource theory, we also show that the bipartite asymptomatic transformation rate obey some distributed relations: when α ≥1,Rα(ρ→ρ') satisfies monogamy relations. Using similar methods, we demonstrate that marginal asymptotic transformation rates and marginal catalytic transformation rates are all satisfies these relations. As a byproduct, we show an equivalence among the asymptomatic transformation rate, marginal asymptotic transformations and marginal catalytic transformations under some restrictions. Here marginal asymptotic transformations and marginal catalytic transformations are special asymptotic transformations, and initial states can be reducible onto target states at nonzero rates. These inequality relations give rise to a new kind of restrictions on the quantum resource distribution and trade off among subsystems. Recently, reversible quantum resource manipulations have been researched, where they have been conjectured that transformations could be executed reversibly in an asymptotic regime. In the future, we will explore a conclusive proof of this conjecture and then study distributions of these reversible manipulations.