@inproceedings{fa5e3e9a5c564ba7a6d952306a1a3714,

title = "Quantum integers",

abstract = "In number theory, a partition of a positive integer n is a way of writing n as a sum of positive integers. The number of partitions of n is given by the partition function p(n). Inspired by quantum information processing, we extend the concept of partitions in number theory as follows: for an integer n, we treat each partition as a basis state of a quantum system representing that number n, so that the Hilbert-space that corresponds to that integer n is of dimension p(n); the {"}classical integer{"} n can thus be generalized into a (pure) quantum state |ψ (n) > which is a superposition of the partitions of n, in the same way that a quantum bit (qubit) is a generalization of a classical bit. More generally,ρ (n) is a density matrix in that same Hilbert-space (a probability distribution over pure states). Inspired by the notion of quantum numbers in quantum theory (such as in Boh{\`r}s model of the atom), we then try to go beyond the partitions, by defining (via recursion) the notion of {"}sub-partitions{"} in number theory. Combining the two notions mentioned above, sub-partitions and quantum integers, we finally provide an alternative definition of the quantum integers [the pure-state |ψ '(n) > and the mixed-state ρ '(n) ], this time using the sub-partitions as the basis states instead of the partitions, for describing the quantum number that corresponds to the integer n.",

keywords = "Density matrix, Integers, Partition, Recursion, Superposition",

author = "Andrei Khrennikov and Moshe Klein and Tal Mor",

year = "2010",

doi = "10.1063/1.3431505",

language = "אנגלית",

isbn = "9780735407770",

series = "AIP Conference Proceedings",

pages = "299--305",

booktitle = "Quantum Theory - Reconsideration of Foundations - 5, QTRF-5",

note = "International Conference Quantum Theory: Reconsideration of Foundations-5, QTRF-5 ; Conference date: 14-06-2009 Through 18-06-2009",

}