Abstract

We consider a class (convex set) of quantum states containing all finite rank states and infinite rank states with the sufficient rate of decreasing of eigenvalues (in particular, all Gaussian states). Quantum states from this class are characterized by the property (called the FA-property) that allows to obtain several results concerning finite-dimensional approximation of basic entropic and information characteristics of quantum systems and channels.

We obtain a simple sufficient condition of the FA-property. We show that all the states with the FA-property form a face of the convex set of all quantum states that is contained within the face of all states with finite von Neumann entropy (the non-coincidence of these two faces follows from the recent result of S. Becker, N. Datta and M.G. Jabbour).

We obtain uniform approximation results for characteristics depending on a pair (channel, input state) and for characteristics depending on a pair (channel, input ensemble). We establish the uniform continuity of the above characteristics as functions of a channel w.r.t. the strong convergence provided that the FA-property holds either for the input state or for the average state of input ensemble.

中文翻译：

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关于具有有限维近似性质的量子态

摘要

我们考虑一类（凸集）量子态，它包含所有有限秩态和无限秩态，具有足够的特征值下降率（特别是所有高斯态）。此类量子态的特征在于允许获得有关量子系统和通道的基本熵和信息特征的有限维近似的几个结果的属性（称为 FA 属性）。

我们得到了 FA 性质的一个简单的充分条件。我们证明所有具有 FA 性质的状态形成所有量子态的凸集的一个面，该面包含在所有具有有限冯诺依曼熵的状态的面内（这两个面的非重合来自最近的结果S. Becker、N. Datta 和 MG Jabbour）。

对于依赖于一对（通道、输入状态）的特性和依赖于一对（通道、输入集合）的特性，我们获得了统一的近似结果。我们将上述特征的统一连续性建立为具有强收敛性的通道的函数，前提是 FA 属性对于输入状态或输入集成的平均状态成立。