The relation between the requirement of efficient implementability and the product-state representation of numbers is examined. Numbers are defined to be any model of the axioms of number theory or arithmetic. Efficient implementability (EI) means that the basic arithmetic operations are physically implementable and the space-time and thermodynamic resources needed to carry out the implementations are polynomial in the range of numbers considered. Different models of numbers are described to show the independence of both EI and the product-state representation from the axioms. The relation between EI and the product-state representation is examined. It is seen that the condition of a product-state representation does not imply EI. Arguments used to refute the converse implication, EI implies a product-state representation, seem reasonable; but they are not conclusive. Thus this implication remains an open question.
- Received 12 April 2001
- Published 12 October 2001
© 2001 The American Physical Society