Ada Chan, Gabriel Coutinho, Whitney Drazen, Or Eisenberg, Chris Godsil, Gabor Lippner, Mark Kempton, Christino Tamon, Hanmeng Zhan
We develop a general spectral framework to analyze quantum fractional revival in quantum spin networks. In particular, we introduce generalizations of the notions of cospectral and strongly cospectral vertices to arbitrary subsets of vertices, and give various examples. This work resolves two open questions of Chan et.~al. ["Quantum Fractional Revival on graphs". Discrete Applied Math, 269:86-98, 2019.]
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Fractional revival, known as a quantum transport phenomenon, is essential for entanglement generation in quantum spin networks. The concept of fractional revival is a generalization of perfect state transfer and periodicity on graphs. In this paper, we propose a sufficient and necessary condition for abelian Cayley graphs having fractional revival between any two distinct vertices. With this characterization, two general constructions of abelian Cayley graphs having fractional revival is presented. Meanwhile, we establish several new families of abelian Cayley graphs admitting fractional reviv…
We develop the theory of fractional revival in the quantum walk on a graph using its Laplacian matrix as the Hamiltonian. We first give a spectral characterization of Laplacian fractional revival, which leads to a polynomial time algorithm to check this phenomenon and find the earliest time when it occurs. We then apply the characterization theorem to special families of graphs. In particular, we show that no tree admits Laplacian fractional revival except for the paths on two and three vertices, and the only graphs on a prime number of vertices that admit Laplacian fractional revival are doub…
Pretty good fractional revival via magnetic fields: theory and examples
本文建立了任意大小子集上的分数共谱性与相当好的量子分数恢复理论，并据此给出磁场诱导该现象的条件及实例。
We initiate the study of approximate quantum fractional revival in graphs, a generalization of pretty good quantum state transfer in graphs. We give a complete characterization of approximate fractional revival in a graph in terms of the eigenvalues and eigenvectors of the adjacency matrix of a graph. This characterization follows from a lemma due to Kronecker on Diophantine approximation, and is similar to the spectral characterization of pretty good state transfer in graphs. Using this, we give a complete characterizations of when approximate fractional revival can occur in paths and in cycl…
Fractional revival occurs between two vertices in a graph if a continuous-time quantum walk unitarily maps the characteristic vector of one vertex to a superposition of the characteristic vectors of the two vertices. This phenomenon is relevant in quantum information in particular for entanglement generation in spin networks. We study fractional revival in graphs whose adjacency matrices belong to the Bose-Mesner algebra of association schemes. A specific focus is a characterization of balanced fractional revival (which corresponds to maximal entanglement) in graphs that belong to the Hamming …
Fractional revival on Cayley graphs over abelian groups
本文研究有限阿贝尔群上凯莱图的分数复兴问题，给出其存在的充要条件，并应用于循环图与立方体类图的刻画。
Fractional revival on semi-Cayley graphs over abelian groups
本文研究了有限阿贝尔群上半凯莱图分数复苏的存在性，给出了充要条件、整数性必要条件及最短时间刻画，并提供了广义二面体群和双循环群上凯莱图的应用实例。
We develop the theory of pretty good fractional revival in quantum walks on graphs using their Laplacian matrices as the Hamiltonian. We classify the paths and the double stars that have Laplacian pretty good fractional revival.
Fractional revival between twin vertices
本文刻画了加权图中孪生顶点在三种矩阵下的分数复兴，并应用于双锥体顶点。
Pretty good fractional revival on Cayley graphs over dicyclic groups
针对双循环群上的 Cayley 图，本文刻画其发生极佳分数复兴的充要条件，并据此给出允许与不允许该现象的充分条件。
Perfect state transfer and fractional revival can be used to move information between pairs of vertices in a quantum network. While perfect state transfer has received a lot of attention, fractional revival is newer and less studied. One problem is to determine the differences between perfect state transfer and fractional revival. If perfect state transfer occurs between two vertices in a graph, the vertices must be cospectral. Further if there is perfect state transfer between vertices $a$ and $b$ in a graph, there cannot be perfect state transfer from $a$ to any other vertex. No examples of …
Fractional revival in complementary prisms of graphs
本文提出一个研究块结构实对称矩阵上分数复兴的通用框架，证明图的互补棱柱在特定条件下具有分数复兴，并刻画了完全图与完全二部图的互补棱柱中的完美对态转移。
Fractional revival is a quantum transport phenomenon important for entanglement generation in spin networks. This takes place whenever a continuous-time quantum walk maps the characteristic vector of a vertex to a superposition of the characteristic vectors of a subset of vertices containing the initial vertex. A main focus will be on the case when the subset has two vertices. We explore necessary and sufficient spectral conditions for graphs to exhibit fractional revival. This provides a characterization of fractional revival in paths and cycles. Our work builds upon the algebraic machinery d…
Fractional revival on quasi-abelian Cayley graphs
针对量子自旋网络中的分数复兴问题，本文刻画连接集为共轭类并集的拟阿贝尔 Cayley 图，给出其发生分数复兴的充要条件，并推广已有阿贝尔群结果。
Quantum fractional revival on unitary Cayley graphs over finite commutative rings
本文研究了有限交换环上的单位Cayley图中量子分数复兴的存在性，并刻画了允许此类复兴的环结构。
Pretty good plus state transfer in cycles
研究图中邻接、拉普拉斯和无向拉普拉斯矩阵的分数复兴，发现特定条件下图补集保持分数复兴，并通过双覆盖建立联系，从而完全描述环及其补图的漂亮良好加状态传输，进而描述加权路径的漂亮良好顶点状态传输。
Quantum fractional revival on zero-divisor graphs over $\mathbb{Z}_n$
研究$\mathbb{Z}_n$零因子图上连续时间量子行走中完美态转移和分数复兴的存在条件，通过等价划分确定顶点对的充分条件，并分析分数复兴在等价划分中的限制。
Fractional revival on oriented Cayley and semi-Cayley graphs over abelian groups
本文研究定向图上的真分数复兴存在性，建立定向图强共谱顶点间真FR的充要条件，证明阿贝尔群上定向Cayley图无真FR，并刻画阿贝尔群上定向半-Cayley图的真FR存在条件。
Quantum Fractional Revival and Entanglement Entropy in Unitary Cayley Graphs
本文扩展了单位ary Cayley 图上量子分数复兴理论，探讨拉普拉斯矩阵哈密顿量下的复兴现象，推导了最小复兴时间公式，并研究了强共谱顶点对与纠缠熵的关系。
State Transfer on Unitary Cayley Graphs and Quadratic Unitary Cayley Graphs
本文定义了幺正凯莱图$X_n$和二次幺正凯莱图$G_n$，分类了允许良好分数状态转移及分数复活的$X_n$，并分类了允许周期性、完美状态转移等性质的$G_n$。
Pretty good fractional revival on abelian Cayley graphs
研究阿贝尔凯莱图上的相当好的分数复兴，通过推导循环图和非循环阿贝尔凯莱图呈现PGFR的充分条件，找到相关图族，还得出部分循环图族呈现PGFR的必要条件，推广了Chan等人关于循环的结果。
We study continuous-time quantum walks on normal Cayley graphs of certain non-abelian groups, called extraspecial groups. By applying general results for graphs in association schemes we determine the precise conditions for perfect state transfer and fractional revival, and use partial spreads to construct graphs on extraspecial $2$-groups admitting these various phenomena. Lastly, we use a result of Ada Chan to show that there is no normal Cayley graph of an extraspecial group that admits instantaneous uniform mixing.
Chromatic polynomials have been studied extensively, giving us results such as the Fundamental Reduction Theorem and closed formulas for the chromatic polynomials of common classes of graphs. Though, none of those extend to the context of fractional colorings. We thus present closed formulas for the "fractional" chromatic polynomial - a function that counts the number of distinct $b$-fold $λ$-colorings on a given graph - of complete graphs, trees and forests, as well as a generalized form of the Fundamental Reduction Theorem.
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