Discrete Quantum Theories

TL;DR

This paper investigates finite-field frameworks for quantum theory, proposing new models that recover many features of conventional quantum mechanics and offer insights into computational resources and measurement costs.

Contribution

It introduces novel finite-field quantum theories with restricted fields that approximate standard quantum mechanics and distinguish between system description and measurement resources.

Findings

Finite-field quantum theories can replicate core quantum structures.

A new framework introduces resource-based measures for system and measurement complexity.

Conventional quantum mechanics emerges as the field size increases in the proposed models.

Abstract

We explore finite-field frameworks for quantum theory and quantum computation. The simplest theory, defined over unrestricted finite fields, is unnaturally strong. A second framework employs only finite fields with no solution to x^2+1=0, and thus permits an elegant complex representation of the extended field by adjoining i=\sqrt{-1}. Quantum theories over these fields recover much of the structure of conventional quantum theory except for the condition that vanishing inner products arise only from null states; unnaturally strong computational power may still occur. Finally, we are led to consider one more framework, with further restrictions on the finite fields, that recovers a local transitive order and a locally-consistent notion of inner product with a new notion of cardinal probability. In this framework, conventional quantum mechanics and quantum computation emerge locally (though not globally) as the size of the underlying field increases. Interestingly, the framework allows one to choose separate finite fields for system description and for measurement: the size of the first field quantifies the resources needed to describe the system and the size of the second quantifies the resources used by the observer. This resource-based perspective potentially provides insights into quantitative measures for actual computational power, the complexity of quantum system definition and evolution, and the independent question of the cost of the measurement process.