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An extension of Chaitin's halting probability Omega to measurement operator in infinite dimensional quantum system | Kohtaro Tadaki
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5 Jul 2004 | Subject: | Quantum Physics; Computational Complexity | quant-ph cs.CC | Abstract: | This paper proposes an extension of Chaitin’s halting probability Omega to measurement operator in 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 infinite dimensional quantum system, called 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. | Source: | arXiv, quant-ph/0407023 | Services: | Forum | Review | PDF | Favorites |
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