量子轨道上Kusuoka测度的遍历性
Ergodicity of Kusuoka measures on quantum trajectories
摘要：
在1989年Kusuoka开始研究的概率测度的转移空间，是确定的帮助下，产品的矩阵。特别是，他推导出了此类测度遍历性的充分条件，这些测度后来被称为Kusuoka测度。我们观察到，对一个酉演化量子系统的重复测量在测量结果序列空间上产生一个Kusuoka测度。我们表明，如果测量包括缩放投影，那么Kusuoka的充分遍历性条件可以显着简化。然后，我们证明了这个条件也是必要的遍历性，如果测量由均匀缩放的秩1投影（即，它是秩-1 POVM），或者正好是两个投影的，其中一个是秩-1。对于后一类的测量，我们还表明，Kusuoka措施是可逆的，在这个意义上，每一个字符串的结果有相同的概率被发射的系统作为其反向。
英文摘要：
In 1989 Kusuoka started the study of probability measures on the shift space that are defined with the help of products of matrices. In particular, he derived a sufficient condition for the ergodicity of such measures, which have since been referred to as Kusuoka measures. We observe that repeated measurements of a unitarily evolving quantum system generate a Kusuoka measure on the space of sequences of measurement outcomes. We show that if the measurement consists of scaled projections, then Kusuoka's sufficient ergodicity condition can be significantly simplified. We then prove that this condition is also necessary for ergodicity if the measurement consists of uniformly scaled rank-1 projections (i.e., it is a rank-1 POVM), or of exactly two projections, one of which is rank-1. For the latter class of measurements we also show that the Kusuoka measure is reversible in the sense that every string of outcomes has the same probability of being emitted by the system as its reverse.
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