An extension of Chaitin's halting probability \Omega to a measurement operator in an infinite dimensional quantum system
Abstract
This paper proposes an extension of Chaitin's halting probability \Omega to a measurement operator in an infinite dimensional quantum system. Chaitin's \Omega is defined as the probability that the universal selfdelimiting 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 programsize 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 socalled universal probability m is introduced first, and then H(s) is defined as log_2 m(s) without reference to the concept of programsize. Mathematically, the statistics of outcomes in a quantum measurement are described by a positive operatorvalued measure (POVM) in the most general setting. Based on the theory of computability structures on a Banach space developed by PourEl and Richards, we extend the universal probability to an analogue of POVM in an infinite dimensional quantum system, called a universal semiPOVM. We also give another characterization of Chaitin's \Omega numbers by universal probabilities. Then, based on this characterization, we propose to define an extension of \Omega as a sum of the POVM elements of a universal semiPOVM. 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 semiPOVM, and study its properties.
 Publication:

arXiv eprints
 Pub Date:
 July 2004
 DOI:
 10.48550/arXiv.quantph/0407023
 arXiv:
 arXiv:quantph/0407023
 Bibcode:
 2004quant.ph..7023T
 Keywords:

 Quantum Physics;
 Computer Science  Computational Complexity
 EPrint:
 24 pages, LaTeX2e, no figures, accepted for publication in Mathematical Logic Quarterly: The title was slightly changed and a section on an operatorvalued algorithmic information theory was added