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arXiv:0808.2261 · Physical Review A

State transfer in highly connected networks and a quantum Babinet principle

Dimitris I. Tsomokos, Martin Bodo Plenio, Inés de Vega, Susana F. Huelga

Published Dec 5, 2008• 18 citations
arXivHTMLPDF

Abstract

The transfer of a quantum state between distant nodes in two-dimensional networks is considered. The fidelity of state transfer is calculated as a function of the number of interactions in networks that are described by regular graphs. It is shown that perfect state transfer is achieved in a network of size $N$, whose structure is that of an $(N∕2)$-cross polytope graph, if $N$ is a multiple of $4$. The result is reminiscent of the Babinet principle of classical optics. A quantum Babinet principle is derived, which allows for the identification of complementary graphs leading to the same fidelity of state transfer, in analogy with complementary screens providing identical diffraction patterns.

DOI
10.1103/physreva.78.062310

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The transfer of a quantum state between distant nodes in two-dimensional networks is considered. The fidelity of state transfer is calculated as a function of the number of interactions in networks that are described by regular graphs. It is shown that perfect state transfer is achieved in a network of size NNN, whose structure is that of an (N∕2)(N∕2)(N∕2)-cross polytope graph, if NNN is a multiple of 444. The result is reminiscent of the Babinet principle of classical optics. A quantum Babinet principle is derived, which allows for the identification of complementary graphs leading to the same fidelity of state transfer, in analogy with complementary screens providing identical diffraction patterns.
Let X be a graph, A its adjacency matrix, and t a non-negative real number. The matrix exp(i t A) determines the evolution in time of a certain quantum system defined on the graph. It represents a continuous-time quantum walk in X. We say that X admits perfect state transfer from a vertex u to a vertex v if there is a time t such that \n| exp(i t A)_(u,v) | = 1. \n \nThe main problem we study in this thesis is that of determining which simple graphs admit perfect state transfer. For some classes of graphs the problem is solved. For example, a path on n vertices admits perfect state transfer if and only if n=2 or n=3. However, the general problem of determining all graphs that admit perfect state transfer is substantially hard. \n \nIn this thesis, we focus on some special cases. We provide necessary and sufficient conditions for a distance-regular graph to admit perfect state transfer. In particular, we provide a detailed account of which distance-regular graphs of diameter three do so. \n \nA graph is said to be spectrally extremal if the number of distinct eigenvalues is equal to the diameter plus one. Distance-regular graphs are examples of such graphs. We study perfect state transfer in spectrally extremal graphs and explore rich connections to the topic of orthogonal polynomials. We characterize perfect state transfer in such graphs. \n \nWe also provide a general framework in which perfect state transfer in graph products can be studied. We use this to determine when direct products and double covers of graphs admit perfect state transfer. As a consequence, we provide many new examples of simple graphs admitting perfect state transfer. We also provide some advances in the understanding of perfect state transfer in Cayley graphs for the groups (Z_2)^d and Z_n. \n \nFinally, we consider the problem of determining which trees admit perfect state transfer. We show more generally that, except for the path on two vertices, if a connected bipartite graph contains a unique perfect matching, then it cannot admit perfect state transfer. We also consider this problem in the context of another model of quantum walks determined by the matrix exp(i t L), where L is the Laplacian matrix of the graph. In particular, we show that no tree on an odd number of vertices admits perfect state transfer according to this model.
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.
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.
We study perfect state transfer on quantum networks represented by weighted graphs. Our focus is on graphs constructed from the join and related graph operators. Some specific results we prove include: (1) The join of a weighted two-vertex graph with any regular graph has perfect state transfer. This generalizes a result of Casaccino et al. [clms09] where the regular graph is a complete graph or a complete graph with a missing link. In contrast, the half-join of a weighted two-vertex graph with any weighted regular graph has no perfect state transfer. This implies that adding weights in a complete bipartite graph do not help in achieving perfect state transfer. (2) A Hamming graph has perfect state transfer between each pair of its vertices. This is obtained using a closure property on weighted Cartesian products of perfect state transfer graphs. Moreover, on the hypercube, we show that perfect state transfer occurs between uniform superpositions on pairs of arbitrary subcubes. This generalizes results of Bernasconi et al. [bgs08] and Moore and Russell [mr02]. Our techniques rely heavily on the spectral properties of graphs built using the join and Cartesian product operators.
Perfect transfer of a quantum state through a one-dimensional chain is now well understood, allowing one not only to decide whether a fixed Hamiltonian achieves perfect transfer but to design a suitable one. We are particularly interested in being able to design, or understand the limitations imposed upon, Hamiltonians subject to various naturally arising constraints such as a limited coupling topology with low connectivity (specified by a graph) and type of interaction. In this paper, we characterize the necessary and sufficient conditions for transfer through a network and describe some natural consequences such as the impossibility of routing between many different recipients for a large class of Hamiltonians and the limitations on transfer rate. We also consider some of the trade-offs that arise in uniformly coupled networks (both Heisenberg and XXXXXX models) between transfer distance and the size of the network as a consequence of the derived conditions.
By using some techniques such as spectral distribution and stratification associated with the graphs, employed in [1,2] for the purpose of Perfect state transfer (PST) of a single qubit over antipodes of distance-regular spin networks and PST of a ddd-level quantum state over antipodes of pseudo-distance regular networks, PST of an m-qubit GHZ state is investigated. To do so, we employ the particular distance-regular networks (called Johnson networks) J(2m,m) to transfer an m-qubit GHZ state initially prepared in an arbitrary node of the network (called the reference node) to the corresponding antipode, perfectly. Keywords: Perfect state transferenc, GHZ states, Johnson network, Stratification, Spectral distribution PACs Index: 01.55.+b, 02.10.Yn