Representation of complex rational numbers in quantum mechanics

A representation of complex rational numbers in quantum mechanics is described that is not based on logical or physical qubits. It stems from noting that the 0’s in a product qubit state do not contribute to the number. They serve only as place holders. The representation is based on the distribution of four types of systems on an integer lattice. The four types, labeled as positive real, negative real, positive imaginary, and negative imaginary, are represented by creation and annihilation operators acting on the system vacuum state. Complex rational number states correspond to products of creation operators acting on the vacuum. Various operators, including those for the basic arithmetic operations, are described. The representation used here is based on occupation number states and is given for bosons and fermions.

Figure 1

(Color online) Example of a nonstandard complex rational state for bosons and fermions for four occupied $j$ values. At each site $j$ the vertical bar shows the occupation numbers (bosons) or extent of $h$ values (fermions) with different shading (colors) showing each of the four types of system. For instance, the bar at site $j-1$ shows $13r+,11r-$, four $i+$, and two $i-$ systems and the bar at site $j$ shows five $r+$ and eight $i+$ bosons. For fermions the ordinate labels the ranges of the $h$ values for each type.