可计算POVM测量中概率的Kolmogorov复杂性上界
Upper bound by Kolmogorov complexity for the probability in computable POVM measurement
摘要：
我们将算法信息论应用于量子力学，以揭示量子力学中固有的算法结构。有两种等价的方法来定义给定的经典有限二进制串s的（经典）柯尔莫哥洛夫复杂度K（s）。在标准方式中，K（s）被定义为通用自定界图灵机输出s的最短输入字符串的长度。另一种方法是先引入所谓的普适概率m，然后定义K（s）为-log_2m（s），而不使用程序大小的概念。我们将普适概率推广到一个矩阵值函数，并将此函数与一个POVM（正算子值测度）相识别。在此基础上，我们研究了有限维量子系统上具有可数测量结果的可计算POVM测量。我们表明，直到乘法常数，2^{-K（s）}是这样的POVM测量中每个测量结果s的概率的上限。在下文中，上界2^{-K（s）}在某种意义上是最优的。
英文摘要：
We apply algorithmic information theory to quantum mechanics in order to shed light on an algorithmic structure which inheres in quantum mechanics. There are two equivalent ways to define the (classical) Kolmogorov complexity K(s) of a given classical finite binary string s. In the standard way, K(s) is defined as the length of the shortest input string for the universal self-delimiting Turing machine to output s. In the other way, we first introduce the so-called universal probability m, and then define K(s) as -log_2 m(s) without using the concept of program-size. We generalize the universal probability to a matrix-valued function, and identify this function with a POVM (positive operator-valued measure). On the basis of this identification, we study a computable POVM measurement with countable measurement outcomes performed upon a finite dimensional quantum system. We show that, up to a multiplicative constant, 2^{-K(s)} is the upper bound for the probability of each measurement outcome s in such a POVM measurement. In what follows, the upper bound 2^{-K(s)} is shown to be optimal in a certain sense.
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