Quantum integers

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 Bohr̀s 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.