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Article overview
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State Transfer in Complex Quantum Walks | Antonio Acuaviva
; Ada Chan
; Summer Eldridge
; Chris Godsil
; Matthew How-Chun-Lun
; Christino Tamon
; Emily Wright
; Xiaohong Zhang
; | Date: |
4 Jan 2023 | 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. | Source: | arXiv, 2301.01473 | Services: | Forum | Review | PDF | Favorites |
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