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在量子资源理论中, 系统中不可避免存在的噪声使得调控和转化量子资源变得困难. 为了克服转化量子资源态时噪声的影响, 高资源初态$ \rho $到低资源目标态$ \rho ' $的转化往往考虑多份初始态到多份目标态的渐近转化. 渐近转化率$ R\left( {\rho \to \rho '} \right) $可以刻画这类转化过程中量子操作的能力, 并且它被定义为目标态份数与初态份数的比值. 一般地, 要得到渐近转化率的确切值是困难的. 在一般的多体量子资源理论中, 本文研究了两部划分下渐近转化率的分布特征: 当$ \alpha \geqslant 1 $时$ {R^\alpha }\left( {\rho \to \rho '} \right) $服从单配性关系式, 并且证明得到边际转化率和边际的催化转化率也都服从上述分布特征. 这些关系式表明多体系统中量子资源的分布以及子系统间量子资源的配置是存在束缚的.In quantum resource theories, manipulating and transforming resource states are often challenging due to the presence of noise. The resource manipulation process from a high resource state $ \rho $ to a low resource state $ \rho ' $ involves asymptotic multiple state replicas, which can be considered as overcoming this problem. Here, the asymptomatic transformation rate $ R\left( {\rho \to \rho '} \right) $ 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, the precise computations of asymptotic transformation rates are challenging, so it is important to establish rigorous and computable boundaries for them. Recently, Ganardi et al. have shown that the transformation rate to any pure state is superadditive for the distillable entanglement. However, it remains a question whether the transformation rate to any noise state is also superadditive in the general resource theory. Firstly, we study the general superadditive inequality satisfied by the transformation rate $ R\left( {\rho \to \rho '} \right) $ of any noise state $ \rho ' $. In any multiple quantum resource theory, we also show that the bipartite asymptomatic transformation rate obeys a distributed relationship: when $ \alpha \geqslant 1 $, $ {R^\alpha }\left( {\rho \to \rho '} \right) $ satisfies monogamy relationship. Using similar methods, we demonstrate that both the marginal asymptotic transformation rate and marginal catalytic transformation rate satisfies these relationships. 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, where the initial state can be reduced into target state at a nonzero rate. These inequality relationships impose a new constraint on the quantum resource distribution and trade off among subsystems. Recently, reversible quantum resource manipulations have been studied, and it is conjectured that transformations can be reversibly executed in an asymptotic regime. In the future, we will explore a conclusive proof of this conjecture and then study the distributions of these reversible manipulations.
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图 1 多体系统中, 两部划分下渐近转化率的单配性关系. 右上图中左灰右白的量子态表示第一部子系统中的态, 右下图中左白右灰的量子态表示第二部子系统中的态
Fig. 1. In any multiple quantum resource theory, the bipartite asymptomatic transformation rates obey monogamy relations. These left gray and right white states in the top-right image are quantum states in the first subsystem, and these left white and right gray states in the below-right image are quantum states in the second subsystem.
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