Representation of natural numbers in quantum mechanics

Abstract

This paper represents one approach to making explicit some of the assumptions and conditions implied in the widespread representation of numbers by composite quantum systems. Any nonempty set and associated operations is a set of natural numbers or a model of arithmetic if the set and operations satisfy the axioms of number theory or arithmetic. This paper is limited to $k\ensuremath{-}\mathrm{ary}$ representations of length L and to the axioms for arithmetic modulo ${k}^{L}.$ A model of the axioms is described based on an abstract L-fold tensor product Hilbert space ${\mathcal{H}}^{\mathrm{arith}}.$ Unitary maps of this space onto a physical parameter based product space ${\mathcal{H}}^{\mathrm{phy}}$ are then described. Each of these maps makes states in ${\mathcal{H}}^{\mathrm{phy}},$ and the induced operators, a model of the axioms. Consequences of the existence of many of these maps are discussed along with the dependence of Grover's and Shor's algorithms on these maps. The importance of the main physical requirement, that the basic arithmetic operations are efficiently implementable, is discussed. This condition states that there exist physically realizable Hamiltonians that can implement the basic arithmetic operations and that the space-time and thermodynamic resources required are polynomial in L.