Locality for quantum systems on graphs depends on the number field

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.

Inverse eigenvalue and related problems for hollow matrices described by graphs

“…Sources of information about nonzero pattern matrices that allow orthogonality include [12,13,14] and the references therein. Much of the literature concerns potentially unitary patterns rather than potentially orthogonal patterns, and it is known that there are patterns that are potentially unitary but not potentially orthogonal [8]. However, few such examples are known, and many of the proofs of results that are stated for potentially unitary work for potentially orthogonal, as is the case with the next result.…”

Inverse eigenvalue and related problems for hollow matrices described by graphs

“…Sources of information about pattern matrices that allow orthogonality include [12,13,14] and the references therein. Much of the literature concerns potentially unitary patterns rather than potentially orthogonal patterns, and it is known that there are patterns that are potentially unitary but not potentially orthogonal [8]. However, few such examples are known, and many of the proofs of results that are stated for potentially unitary work for potentially orthogonal, as is the case with the next result.…”