An extension of Chaitin's halting probability Ωto a measurement operator in an infinite dimensional quantum system
| dc.creator | Tadaki, Kohtaro | |
| dc.date | 2004-07-05 | |
| dc.date | 2006-07-13 | |
| dc.date.accessioned | 2026-07-07T06:40:42Z | |
| dc.date.available | 2026-07-07T06:40:42Z | |
| dc.description | 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. | |
| dc.description | 24 pages, LaTeX2e, no figures, accepted for publication in Mathematical Logic Quarterly: The title was slightly changed and a section on an operator-valued algorithmic information theory was added | |
| dc.identifier | https://arxiv.org/abs/quant-ph/0407023 | |
| dc.identifier | http://arxiv.org/abs/quant-ph/0407023 | |
| dc.identifier | Mathematical Logic Quarterly, Vol.52, 419-438 (2006) | |
| dc.identifier | doi:10.1002/malq.200410061 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/101462 | |
| dc.subject | Quantum Physics | |
| dc.subject | Computational Complexity | |
| dc.title | An extension of Chaitin's halting probability Ωto a measurement operator in an infinite dimensional quantum system | |
| dc.type | text |