凸态体与凸映射
Convex bodies of states and maps
摘要：
当有限维Hilbert空间上的量子态轨道在紧群的幺正作用下的凸包在所有密度态的周围空间中具有非空内部时，我们给出了这个问题的一般解.同样的方法可以应用于研究量子信道的凸组合。这两个问题的重要性源于这样一个事实，即通常只有在所有状态或通道的嵌入空间中具有非零体积的集合才具有实际重要性。对于二分系统上的局域变换群，我们用通过它们的轨道凸包的性质来刻画最大纠缠态。我们还比较了凸体的最大球和最大体积椭球方面的两个部分特征，并表明，在一般情况下，它们不重合。可分离态、混合幺正通道和k纠缠态也被认为是我们技术的例子。
英文摘要：
We give a general solution to the question when the convex hulls of orbits of quantum states on a finite-dimensional Hilbert space under unitary actions of a compact group have a non-empty interior in the surrounding space of all density states. The same approach can be applied to study convex combinations of quantum channels. The importance of both problems stems from the fact that, usually, only sets with non-vanishing volumes in the embedding spaces of all states or channels are of practical importance. For the group of local transformations on a bipartite system we characterize maximally entangled states by properties of a convex hull of orbits through them. We also compare two partial characteristics of convex bodies in terms of largest balls and maximum volume ellipsoids contained in them and show that, in general, they do not coincide. Separable states, mixed-unitary channels and k-entangled states are also considered as examples of our techniques.
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