State Transfer in Complex Quantum Walks

TL;DR

This work analyzes state transfer in quantum walks on graphs with Hermitian adjacency, unifying PST and PGST phenomena across oriented and Hermitian graphs. It proves that universal PST in oriented graphs is exceptional, occurring only for $K_2$ and the oriented $3$-cycle, and it constructs infinite families with PST on a four-vertex subset and infinite families exhibiting one-way PST without periodicity. It further develops a theory for PST in Hermitian graphs, showing phase algebraicity under algebraic spectra and providing one-way PST constructions via transcendental data. Finally, it demonstrates infinite families of pretty good state transfer in rooted products, including oriented 3-cycles rooted with stars and circulants rooted with looped paths, using a blend of spectral methods, orthogonal polynomials, and field-trace arguments. The results generalize known examples (e.g., double stars and looped paths) and connect to experimental observations of quantum transport and information routing on networks.

Abstract

Given a graph with Hermitian adjacency matrix $H$, perfect state transfer occurs from vertex $a$ to vertex $b$ if the $(b,a)$-entry of the unitary matrix $\exp(-iHt)$ has unit magnitude for some time $t$. This phenomenon is relevant for information transmission in quantum spin networks and is known to be monogamous under real symmetric matrices. We prove the following results: 1. For oriented graphs (whose nonzero weights are $\pm i$), the oriented $3$-cycle and the oriented edge are the only graphs where perfect state transfer occurs between every pair of vertices. This settles a conjecture of Cameron et al. On the other hand, we construct an infinite family of oriented graphs with perfect state transfer between any pair of vertices on a subset of size four. 2. There are infinite families of Hermitian graphs with one-way perfect state transfer, where perfect state transfer occurs without periodicity. In contrast, perfect state transfer implies periodicity whenever the adjacency matrix has algebraic entries (as shown by Godsil). 3. There are infinite families with non-monogamous pretty good state transfer in rooted graph products. In particular, we generalize known results on double stars (due to Fan and Godsil) and on paths with loops (due to Kempton, Lippner and Yau). The latter extends the experimental observation of quantum transport (made by Zimborás et al.) and shows non-monogamous pretty good state transfer can occur amongst distant vertices.

Theorems & Definitions (43)

• Theorem 2.1
• proof
• Theorem 2.2
• proof
• Theorem 2.3
• Theorem 2.4
• proof
• Theorem 3.1
• Lemma 3.2
• proof