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Linear and Multilinear Algebra

A characterization of state transfer on double subdivided stars

Sarojini Mohapatra, Hiranmoy Pal

Published Feb 14, 2025• 0 citations
Publisher

Abstract

A subdivided star SK1,l is obtained by identifying exactly one pendant vertex from l copies of the path P3. We consider the existence of quantum state transfer on double subdivided star Tl,m which is a pair of subdivided stars SK1,l and SK1,m joined by an edge to the respective coalescence vertices. We provide a complete characterization of double subdivided stars exhibiting pretty good state transfer. Although there is no perfect state transfer on double subdivided stars, we find perfect pair state transfer on Tl,m considering the adjacency matrix as the Hamiltonian of the corresponding quantum system.

DOI
10.1080/03081087.2025.2464657

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A subdivided star SK1,l is obtained by identifying exactly one pendant vertex from l copies of the path P3. We consider the existence of quantum state transfer on double subdivided star Tl,m which is a pair of subdivided stars SK1,l and SK1,m joined by an edge to the respective coalescence vertices. We provide a complete characterization of double subdivided stars exhibiting pretty good state transfer. Although there is no perfect state transfer on double subdivided stars, we find perfect pair state transfer on Tl,m considering the adjacency matrix as the Hamiltonian of the corresponding quantum system.
We give a complete characterization of pretty good state transfer on paths between any pair of vertices with respect to the quantum walk model determined by the XY-Hamiltonian. If nnn is the length of the path, and the vertices are indexed by the positive integers from 1 to nnn, with adjacent vertices having consecutive indices, then the necessary and sufficient conditions for pretty good state transfer between vertices aaa and bbb are that (a) a+b=n+1a + b = n + 1a+b=n+1, (b) n+1n + 1n+1 has at most one odd non-trivial divisor, and (c) if n=2tr−1n = 2^t r - 1n=2tr−1, for rrr odd and r≠1r \neq 1r=1, then aaa is a multiple of 2t−12^{t - 1}2t−1.
We study the existence of quantum state transfer in Q\mathcal{Q}Q-graphs in this paper. The Q\mathcal{Q}Q-graph of a graph GGG, denoted by Q(G)\mathcal{Q}(G)Q(G), is the graph derived from GGG by plugging a new vertex to each edge of GGG and joining two new vertices which lie on adjacent edges of GGG by an edge. We show that, if all eigenvalues of a regular graph GGG are integers, then its Q\mathcal{Q}Q-graph Q(G)\mathcal{Q}(G)Q(G) has no perfect state transfer. In contrast, we also prove that the Q\mathcal{Q}Q-graph of a regular graph has pretty good state transfer under some mild conditions. Finally, applying the obtained results, we also exhibit many new families of Q\mathcal{Q}Q-graphs having no perfect state transfer, but admitting pretty good state transfer.
We give a complete characterization of pretty good state transfer on paths between any pair of vertices with respect to the quantum walk model determined by the XY-Hamiltonian. If n is the length of the path, and the vertices are indexed by the positive integers from 1 to n, with adjacent vertices having consecutive indices, then the necessary and sufficient conditions for pretty good state transfer between vertices a and b are that (a) a + b = n + 1, (b) n + 1 has at most one positive odd non-trivial divisor, and (c) if n = 2^t r - 1, for r odd and r \neq 1, then a is a multiple of 2^{t - 1}.
Given a graph with Hermitian adjacency matrix HHH, perfect state transfer occurs from vertex aaa to vertex bbb if the (b,a)(b,a)(b,a)-entry of the unitary matrix exp⁡(−iHt)\exp(-iHt)exp(−iHt) has unit magnitude for some time ttt. 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 ±i\pm i±i), the oriented 333-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.
Let G be a graph and let t be a positive real number. Then the evolution of the continuous quantum walk defined on G is described by the transition matrix U(t)=exp(itH).The matrix H is called Hamiltonian. So far the most studied quantum walks are the ones whose Hamiltonians are the adjacency matrices of the underlying graphs and initial states are vertex states e_a, with e_a being the characteristic vector of vertex a. \n \nThis thesis focuses on Laplacian edge state transfer, that is, the quantum walks whose initial states are edge states e_a-e_b and Hamiltonians are the Laplacians of the underlying graphs. So far the research about perfect state transfer only involves vertex states and Laplacian edge state transfer has not been studied before. We extend the known results of perfect vertex state transfer and periodicity of vertex states to Laplacian edge state transfer. \n \nWe prove two useful closure properties for perfect Laplacian edge state transfer. One is that complementation preserves perfect edge state transfer. The other is that if G has perfect Laplacian edge state transfer at time τ and H has perfect Laplacian vertex state transfer also at time τ, then with some mild assumption on the pairs of vertex states and edge states that have perfect state transfer, the Cartesian product G □ H also admits perfect edge state transfer. Those two properties provide us new ways to construct graphs with perfect Laplacian edge state transfer. We also observe one phenomenon that happens in Laplacian edge state transfer which never happens in vertex state transfer: if there is perfect state transfer from e_a-e_b to e_α-e_β and also from e_b-e_c to e_β-e_γ at the same time t in G, then G admits perfect state transfer from e_a-e_c to e_α-e_γ at time t. \n \nWe give characterizations of perfect Laplacian edge state transfer in cycles, paths and complete bipartite graphs K_{2,4n}. We study perfect state transfer and periodicity on edge states with special spectral features. We also consider the case when the unsigned Laplacian is Hamiltonian and initial states are plus states of the form e_a+e_b. In this case, we characterize perfect state transfer in paths, cycles and bipartite graphs. \nWe close this thesis by a list of open questions.
The neighborhood corona G⋆HG \star HG⋆H is the graph obtained by taking one copy GGG and ∣G∣|G|∣G∣ copies of HHH, and joining each vertex of the jjjth copy of HHH to all neighbors of vjv_{j}vj​ in GGG. In this paper, we study the state transfer of neighborhood coronas related to the adjacency matrix. Concretely, we provide some necessary conditions under which the neighborhood corona G⋆HG \star HG⋆H admits perfect state transfer, and obtain a new family of graphs with pretty good state transfer based on neighborhood coronas.