Locality for quantum systems on graphs depends on the number field

TL;DR

This paper investigates how the choice of number field (real, complex, or quaternionic) affects the existence of quantum dynamics on directed graphs, revealing that complex numbers provide the broadest support for such processes.

Contribution

It characterizes the digraphs that admit saturated Z-local quantum dynamics over different number fields, highlighting the unique role of complex numbers in quantum mechanics.

Findings

Complex amplitudes allow more digraphs for quantum dynamics than real or rational numbers.

A construction distinguishes complex from quaternionic quantum mechanics.

Complex numbers support the largest class of topologies for discrete quantum evolution.

Abstract

Adapting a definition of Aaronson and Ambainis [Theory Comput. 1 (2005), 47--79], we call a quantum dynamics on a digraph "saturated Z-local" if the nonzero transition amplitudes specifying the unitary evolution are in exact correspondence with the directed edges (including loops) of the digraph. This idea appears recurrently in a variety of contexts including angular momentum, quantum chaos, and combinatorial matrix theory. Complete characterization of the digraph properties that allow such a process to exist is a long-standing open question that can also be formulated in terms of minimum rank problems. We prove that saturated Z-local dynamics involving complex amplitudes occur on a proper superset of the digraphs that allow restriction to the real numbers or, even further, the rationals. Consequently, among these fields, complex numbers guarantee the largest possible choice of topologies supporting a discrete quantum evolution. A similar construction separates complex numbers from the skew field of quaternions. The result proposes a concrete ground for distinguishing between complex and quaternionic quantum mechanics.