Efficient Implementation and the Product State Representation of Numbers

In all physical representations of numbers constructed to date, numbers are represented by strings of numerals or by tensor product states of systems in quantum mechanics. This is the case for macroscopic systems, such as classical computers, which are in such wide use. It is also true for microscopic systems or quantum computers which are of much recent interest [1,2]. The universal use of these representations brings up the question, are these string or tensor product state representations necessary? Or is it just a matter of convenience rather than necessity that representations constructed to date have this property? This question will be examined here by studying physical models of the axioms for number theory. Since these axioms are supposed to describe natural numbers (the nonnegative integers), it follows that any physical model of the axioms is a physical model of the natural numbers. Since the (nonlogical) axioms of number theory are referred to often, it is worth stating them explicitly. In