Physics – Quantum Physics
Scientific paper
2004-07-05
Mathematical Logic Quarterly, Vol.52, 419-438 (2006)
Physics
Quantum Physics
24 pages, LaTeX2e, no figures, accepted for publication in Mathematical Logic Quarterly: The title was slightly changed and a
Scientific paper
10.1002/malq.200410061
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.
No associations
LandOfFree
An extension of Chaitin's halting probability Ωto a measurement operator in an infinite dimensional quantum system does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with An extension of Chaitin's halting probability Ωto a measurement operator in an infinite dimensional quantum system, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and An extension of Chaitin's halting probability Ωto a measurement operator in an infinite dimensional quantum system will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-73049