Transport across quantum networks underlies many problems, from state transfer on a spin network to energy transport in photosynthetic complexes. However, networks can contain dark subspaces that block the transportation, and various methods used to enhance transfer on quantum networks can be viewed as equivalently avoiding, modifying, or destroying the dark subspace. Here, we exploit graph theoretical tools to identify the dark subspaces and show that asymptotically almost surely they do not exist for large networks, while for small ones they can be suppressed by properly perturbing the coupling rates between the network nodes. More specifically, we apply these results to describe the recently experimentally observed and robust transport behaviour of the electronic excitation travelling on a genetically-engineered light-harvesting cylinder (M13 virus) structure. We believe that these mainly topological tools may allow us to better infer which network structures and dynamics are more favourable to enhance transfer of energy and information towards novel quantum technologies.
We consider a network of n spin 12 systems which are pairwise interacting via Ising interaction and are controlled by the same electro-magnetic control field. Such a system presents symmetries since the Hamiltonian is unchanged if we permute two spins. This prevents full (operator) controllability, in that not every unitary evolution can be obtained. We prove however that controllability is verified if we restrict ourselves to unitary evolutions which preserve the above permutation invariance. For low dimensional cases, n = 2 and n = 3, we provide an analysis of the Lie group of available evolutions and give explicit control laws to transfer between two arbitrary permutation invariant states. This class of states includes highly entangled states such as Greenberger-Horne-Zeilinger (GHZ) states and W states, which are of interest in quantum information.
Much work has been done in the last two decades on the topic of quantum state transfer in a quantum spin network. One can model such a system of interacting qubits using an undirected graph, and studying vertex-to-vertex dynamics. This setup has recently been relaxed to allow for dynamics between linear combinations of two vertex states, i.e.\ from
u=ea+seb to
μ=eα+reβ, where
r=s is either
−1 (which corresponds to pair state transfer) or
+1 (which corresponds to plus state transfer), or more recently
r=s is taken to be any real number (which corresponds to
s-pair state transfer).
Here, we broaden the investigation of
s-pair state transfer to \textit{perfect
(s,r)-state transfer}, which is perfect state transfer from
u=ea+seb to
μ=eα+reβ (up to some dilation) where
r,s∈C. We identify infinite families of graphs with perfect
(s,r)-state transfer and provide characterizations of cases when
∣r∣=∣s∣ and when
∣r∣=∣s∣, showing situations when the degree of entanglement between vertex states is preserved, and when it is not preserved. The latter is particularly important as it represents perfect state transfer from an entangled pair of qubits to another one where the degree of entanglement need not be the same\mdash in fact, it can be set up so as to ``boost'' (increase) entanglement. We provide an algorithm that finds the vector with two nonzero entries that maximizes the fidelity of transfer for a fixed time
t starting from a given
s-pair state
u. Finally, we provide a sensitivity analysis, with respect to readout time errors, of perfect
(s,r)-state transfer.
Reliably transferring a quantum state from one location to another, as well as generating entangled states, are important tasks to achieve in quantum spin systems. The fidelity or probability of state transfer is a number between 0 and 1 that measures the closeness of two quantum states. Fidelity is used to determine the accuracy of quantum state transfer. There are several interesting phenomena of quantum state transfer defined via fidelity: perfect state transfer, pretty good state transfer, and fractional revival. This thesis contains results about the perfect state transfer property of some special classes of graphs, including Hadamard diagonalizable graphs, weighted paths with loops, as well as switched and partially switched hypercubes. A correspondence between the class of graphs that are diagonalizable by a standard Hadamard matrix and the class of cubelike graphs is given. Sensitivity of fidelity to errors when perfect state transfer occurs is analysed: if a system admits perfect state transfer at some time t, bounds on fidelity of state transfer at t+h for very small h are given, as well as bounds on fidelity of a slightly perturbed system at time t. Finally, Laplacian fractional revival on graphs is considered; in particular the thesis contains a characterization of threshold graphs that admit Laplacian fractional revival.
Quantum spin systems carry information in the form of quantum states, and the propagation of these quantum states is described by a quantum walk. In order to construct an operational quantum computer, the task of reliably transmitting quantum states from one part of a quantum computer to another must be accomplished. Quantum walks are known to be powerful tools in achieving this task. Pure states are quantum states represented by unit complex vectors. In this thesis, we use algebraic and combinatorial techniques to develop the theory of perfect state transfer on pure states in quantum walks, with emphasis on the adjacency and Laplacian matrices as Hamiltonians of a graph representing a spin network. We prove basic results about eigenvalue supports, periodicity, and strong cospectrality on pure states. We investigate a special type of strong cospectrality called m-strong cospectrality, and show that m-strongly cospectral pure states admit desirable quantum state transfer properties. Several characterisations of perfect state transfer are also given, one for general pure states, one for m-strongly pure states, and another one for real pure states. These characterisations give rise to algebraic and analytic properties of pure states admitting perfect state transfer. Moreover, we determine all complete graphs, complete bipartite graphs, cycles, and paths that admit perfect state transfer between m-strongly pure states. Constructions of infinite families of graphs admitting perfect state transfer are also given. We also characterise perfect state transfer between vertex states in joins and blowup graphs. We determine when the join operation preserves or induces perfect state transfer in the resulting graph. We also use blow-up graphs to construct new families of regular graphs with perfect state transfer. We use these two graph operations to demonstrate that graphs with perfect state transfer can be constructed from graphs that do not exhibit such a property. Finally, we investigate perfect state transfer on s-pair states – pure states that represent entanglement between two vertices. We establish combinatorial and spectral properties of s-pair states with perfect state transfer, and characterise its existence in complete graphs, complete bipartite graphs, paths and distance-regular graphs. Further, we construct infinite families of graphs with s-pair state transfer.
Highly efficient transfer of quantum resources including quantum excitations, states, and information on coupled bosonic networks is an important task in quantum physics and quantum information science. Here we propose a bipartite-graph framework to characterize quantum excitation transfer in coupled bosonic networks. This is achieved by diagonalizing the intermediate subnetwork between the sender and the receiver to construct a bipartite-graph configuration, and hence this treatment can be understood from the viewpoint of network deformation. We examine the covariance matrix of the coupled bosonic networks in both the original and bipartite-graph representations. In particular, we investigate quantum excitation transfer in both the finite and infinite intermediate-normal-mode cases and show the dependence of the transfer efficiency on the network configurations and system parameters. We find the bounds of maximally transferred excitations for various network configurations and reveal the underlying physical mechanisms. We also discover that the dark-mode effect will degrade the excitation transfer efficiency. Our findings provide a new insight for the design and optimization of networks in physics, information theory, and complex system science.