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An extension of Chaitin's halting probability Ω to a measurement operator in an infinite dimensional quantum system
Author(s) -
Tadaki Kohtaro
Publication year - 2006
Publication title -
mathematical logic quarterly
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.473
H-Index - 28
eISSN - 1521-3870
pISSN - 0942-5616
DOI - 10.1002/malq.200410061
Subject(s) - povm , mathematics , discrete mathematics , hilbert space , measure (data warehouse) , operator (biology) , probability measure , string (physics) , pure mathematics , quantum , quantum operation , computer science , quantum mechanics , biochemistry , chemistry , physics , database , repressor , open quantum system , transcription factor , mathematical physics , gene
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 –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 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 $ \hat H $ ( 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)