An extension of Chaitin's halting probability \Omega to a measurement operator in an infinite dimensional quantum system
<|reference_start|>An extension of Chaitin's halting probability \Omega to a measurement operator in an infinite dimensional quantum system: 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 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 \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 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.<|reference_end|>
@article{tadaki2004an,
    title={An extension of Chaitin's halting probability \Omega to a measurement
  operator in an infinite dimensional quantum system},
    author={Kohtaro Tadaki},
    journal={Mathematical Logic Quarterly, Vol.52, 419-438 (2006)},
    year={2004},
    doi={10.1002/malq.200410061},
    archivePrefix={arXiv},
    eprint={quant-ph/0407023},
    primaryClass={quant-ph cs.CC}
}