A further look into combinatorial orthogonality

Strongly quadrangular matrices have been introduced in the study of the combinatorial properties of unitary matrices. It is known that if a (0, 1)-matrix supports a unitary then it is strongly quadrangular. However, the converse is not necessarily true. In this paper, we fully classify strongly quadrangular matrices up to degree 5. We prove that the smallest strongly quadrangular matrices which do not support unitaries have exactly degree 5. Further, we isolate two submatrices not allowing a (0, 1)-matrix to support unitaries.Comment: 11 pages, some typos are corrected. To appear in The Electronic journal of Linear Algebr

Rational Orthogonal versus Real Orthogonal

The main question we raise here is the following one: given a real orthogonal n by n matrix X, is it true that there exists a rational orthogonal matrix Y having the same zero-pattern? We conjecture that this is the case and prove it for n<=5. We also consider the related problem for symmetric orthogonal matrices.Comment: 23 page

Locality for quantum systems on graphs depends on the number field

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.Comment: 9 page

Combinatorially Orthogonal Paths

Vectors x=(x1,x2,...,xn)T and y=(y1,y2,...,yn)T are combinatorially orthogonal if |{i:xiyi≠0}|≠1. An undirected graph G=(V,E) is a combinatorially orthogonal graph if there exists f:V→ℝn such that for any u,v∈V, uv∉E iff f(u) and f(v) are combinatorially orthogonal. We will show that every graph has a combinatorially orthogonal representation. We will show the bounds for the combinatorially orthogonal dimension of any path Pn