Chaitin停驻概率Ω对无限维量子系统中测量算符的推广
An extension of Chaitin's halting probability Ωto a measurement operator in an infinite dimensional quantum system
摘要：
本文将Chaitin的停止概率Ω推广到无限维量子系统中的测量算符。Chaitin的Ω被定义为通用自定界图灵机U停止的概率，在算法信息论的发展中起着核心作用。在理论上，有两种等价的方法来定义给定的有限二进制串s的程序大小复杂度H（s）。在标准方式中，H（s）被定义为U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，然后在不考虑程序大小的情况下定义H（s）为-log_2m（s）。在数学上，量子测量中的结果的统计在最一般的设置中由正算子值测量（POVM）描述。基于Pour-El和Richards发展的Banach空间上的可计算结构理论，我们将普适概率扩展到无限维量子系统中的POVM模拟，称为普适半POVM。我们还利用普适概率给出了Chaitin Ω数的另一个刻画。然后，基于这个特征，我们提出定义Ω的扩张为泛半POVM的POVM元素之和。讨论了这一定义的有效性。在下文中，我们使用泛半POVM在无限维Hilbert空间中引入H（s）的算子版本\hat{H}（s），并研究其性质。
英文摘要：
This paper proposes an extension of Chaitin's halting probability Ωto a measurement operator in an infinite dimensional quantum system. Chaitin's Ωis defined as the probability that the universal self-delimiting Turing machine U halts, and plays a central role in the development of algorithmic information theory. In the theory, there are two equivalent ways to define the program-size complexity H(s) of a given finite binary string s. In the standard way, H(s) is defined as the length of the shortest input string for U to output s. In the other way, the so-called universal probability m is introduced first, and then H(s) is defined as -log_2 m(s) without reference to the concept of program-size. Mathematically, the statistics of outcomes in a quantum measurement are described by a positive operator-valued measure (POVM) in the most general setting. Based on the theory of computability structures on a Banach space developed by Pour-El and Richards, we extend the universal probability to an analogue of POVM in an infinite dimensional quantum system, called a universal semi-POVM. We also give another characterization of Chaitin's Ωnumbers by universal probabilities. Then, based on this characterization, we propose to define an extension of Ωas a sum of the POVM elements of a universal semi-POVM. The validity of this definition is discussed. In what follows, we introduce an operator version \hat{H}(s) of H(s) in a Hilbert space of infinite dimension using a universal semi-POVM, and study its properties.
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