An extension of Chaitin's halting probability Ω to a measurement operator in an infinite dimensional quantum system.

This paper proposes an extension of Chaitin's halting probability Ω to a measurement operator in an infinite dimensional quantum system. Chaitin's Ω is 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 –log₂m (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 Ω numbers by universal probabilities. Then, based on this characterization, we propose to define an extension of Ω as 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 <UEQN>$ \hat H $</UEQN>(s) of H (s) in a Hilbert space of infinite dimension using a universal semi-POVM, and study its properties. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [ABSTRACT FROM AUTHOR]