Quantum state transfer between twins in graphs
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Date
2021-08
Authors
Monterde, Hermie
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Abstract
A quantum spin network can be modelled by an undirected graph whose vertices
and edges represent qubits and their interactions in the network, respectively. One
major problem involving quantum spin networks is determining a time such that
the state at one vertex is transported to another vertex at time with a particular
level of probability, called the fidelity of quantum state transfer. Various useful types
of quantum state transfer arise depending on the fidelity: periodicity, perfect state
transfer, pretty good state transfer, and fractional revival.
In this thesis, we investigate quantum state transfer between twin vertices in
weighted graphs with or without loops. We provide spectral and algebraic characterizations
of twin vertices, and use these to identify which properties of twin vertices
are essential for various types of quantum state transfer to occur between any two of
them. A characterization of vertices in unweighted graphs that exhibit strong cospectrality
with an additional vertex in the post-twinning graph is also given. Moreover,
we determine necessary and sufficient conditions for periodicity, perfect state transfer,
pretty good state transfer, and fractional revival to occur between twin vertices
with respect to adjacency dynamics. Then, we apply our results to double cones on
regular graphs, which are a special class of graphs with twin vertices. Finally, we
explore quantum state transfer in some common families of graphs with respect to
adjacency dynamics.
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Keywords
Twin vertices, Quantum state transfer, Quantum walks, Twinning