arXiv subjects

Stephen Kirkland

Publications and source records attributed to Stephen Kirkland.

14 recordsLinked to original sources

arXiv2026

Perfect $(s,r)$-state transfer

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 $\mathbf u = \mathbf e_a + s \mathbf e_b$ to $\mathbf \mu=\mathbf e_{\alpha} + r \mathbf e_{\beta}$, 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 $\mathbf u = \mathbf e_a + s \mathbf e_b$ to $\mathbf \mu=\mathbf e_{\alpha} + r \mathbf e_{\beta}$ (up to some dilation) where $r,s\in \mathbb C$. We identify infinite families of graphs with perfect $(s,r)$-state transfer and provide characterizations of cases when $|r|= |s|$ and when $|r|\neq |s|$, 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 $\mathbf u$. Finally, we provide a sensitivity analysis, with respect to readout time errors, of perfect $(s,r)$-state transfer.

quant-ph↗

arXiv2025

Perfect state transfer between real pure states

Pure states correspond to one-dimensional subspaces of $\mathbb{C}^n$ represented by unit vectors. In this paper, we develop the theory of perfect state transfer (PST) between real pure states with emphasis on the adjacency and Laplacian matrices as Hamiltonians of a graph representing a quantum spin network. We characterize PST between real pure states based on the spectral information of a graph and prove three fundamental results: (i) every periodic real pure state $\mathbf{x}$ admits perfect state transfer with another real pure state $\mathbf{y}$, (ii) every connected graph admits perfect state transfer between real pure states, and (iii) for any pair of real pure states $\mathbf{x}$ and $\mathbf{y}$ and for any time $\tau$, there exists a real symmetric matrix $M$ such that $\mathbf{x}$ and $\mathbf{y}$ admits perfect state transfer relative to $M$ at time $\tau$. We also determine all real pure states that admit PST in complete graphs, complete bipartite graphs, paths, and cycles. This leads to a complete characterization of pair and plus state transfer in paths and complete bipartite graphs. We give constructions of graphs that admit PST between real pure states. Finally, using results on the spread of graphs, we prove that amongst all $n$-vertex simple unweighted graphs, the least minimum PST time between real pure states relative to the Laplacian is attained by any join graph, while the it is attained by the join of an empty graph and a complete graph of appropriate sizes relative to the adjacency matrix.

quant-ph↗

arXiv2024

A generalization of quantum pair state transfer

An $s$-pair state in a graph is a quantum state of the form $\mathbf{e}_u+s\mathbf{e}_v$, where $u$ and $v$ are vertices in the graph and $s$ is a non-zero complex number. If $s=-1$ (resp., $s=1$), then such a state is called a pair state (resp. plus state). In this paper, we develop the theory of perfect $s$-pair state transfer in continuous quantum walks, where the Hamiltonian is taken to be the adjacency, Laplacian or signless Laplacian matrix of the graph. We characterize perfect $s$-pair state transfer in complete graphs, cycles and antipodal distance-regular graphs admitting vertex perfect state transfer. We construct infinite families of graphs with perfect $s$-pair state transfer using quotient graphs and graphs that admit fractional revival. We provide necessary and sufficient conditions such that perfect state transfer between vertices in the line graph relative to the adjacency matrix is equivalent to perfect state transfer between the plus states formed by corresponding edges in the graph relative to the signless Laplacian matrix. Finally, we characterize perfect state transfer between vertices in the line graphs of Cartesian products relative to the adjacency matrix.

quant-ph↗

arXiv2024

State Transfer and Readout Times for Trees of Diameter 4

We consider the state transfer properties of continuous time quantum walks on trees of diameter 4. We characterize all pairs of strongly cospectral vertices in trees of diameter 4, finding that they fall into pairs of three different types. For each type, we construct an infinite family of diameter 4 trees for which there is pretty good state transfer between the pair of strongly cospectral vertices. Moreover, for two of those types, for each tree in the infinite family, we give an explicit sequence of readout times at which the fidelity of state transfer converges to $1$. For strongly cospectral vertices of the remaining type, we identify a sequence of trees and explicit readout times so that the fidelity of state transfer between the strongly cospectral vertices approaches $1.$ We also prove a result of independent interest: for a graph with the property that the fidelity of state transfer between a pair of vertices at time $t_k$ converges to $1$ as $k \rightarrow \infty,$ then the derivative of the fidelity at $t_k$ converges to $0$ as $k \rightarrow \infty. $

quant-ph↗

arXiv2023

Bounds on Kemeny's constant of a graph and the Nordhaus-Gaddum problem

We study Nordhaus-Gaddum problems for Kemeny's constant $\mathcal{K}(G)$ of a connected graph $G$. We prove bounds on $\min\{\mathcal{K}(G),\mathcal{K}(\overline{G})\}$ and the product $\mathcal{K}(G)\mathcal{K}(\overline{G})$ for various families of graphs. In particular, we show that if the maximum degree of a graph $G$ on $n$ vertices is $n-O(1)$ or $n-Ω(n)$, then $\min\{\mathcal{K}(G),\mathcal{K}(\overline{G})\}$ is at most $O(n)$.

math.CO↗

arXiv2023

Completion Problems and Sparsity for Kemeny's Constant

For a partially specified stochastic matrix, we consider the problem of completing it so as to minimize Kemeny's constant. We prove that for any partially specified stochastic matrix for which the problem is well-defined, there is a minimizing completion that is as sparse as possible. We also find the minimum value of Kemeny's constant in two special cases: when the diagonal has been specified, and when all specified entries lie in a common row.

math.SP↗

arXiv2023

Powers of Karpelevic arcs and their Sparsest Realising matrices

The region in the complex plane containing the eigenvalues of all stochastic matrices of order n was described by Karpelevic in 1988, and it is since then known as the Karpelevic region. The boundary of the Karpelevic region is the union of disjoint arcs called the Karpelevic arcs. We provide a complete characterization of the Karpelevic arcs that are powers of some other Karpelevic arc. Furthermore, we find the necessary and sufficient conditions for a sparsest stochastic matrix associated with the Karpelevic arc of order n to be a power of another stochastic matrix.

math.CO↗

arXiv2023

Edge Addition and the Change in Kemeny's Constant

Given a connected graph $G$, Kemeny's constant $\mathcal{K}({G})$ measures the average travel time for a random walk to reach a randomly selected vertex. It is known that when an edge is added to $G$, the value of Kemeny's constant may either decrease, increase, or stay the same. In this paper, we present a quantitative analysis of this behaviour when the initial graph is a tree with $n$ vertices. We prove that when an edge is added into a tree on $n$ vertices, the maximum possible increase in Kemeny's constant is roughly $\frac{2}{3}n,$ while the maximum possible decrease is roughly $\frac{3}{16}n^2$. We also identify the trees, and the edges to be added, that correspond to the maximum increase and maximum decrease. Throughout, both matrix theoretic and graph theoretic techniques are employed.

math.CO↗

arXiv2023

Quantum state transfer between twins in weighted graphs

Twin vertices in simple unweighted graphs are vertices that have the same neighbours and, in the case of weighted graphs with possible loops, the corresponding incident edges have equal weights. In this paper, we explore the role of twin vertices in quantum state transfer. In particular, we provide characterizations of periodicity, perfect state transfer, and pretty good state transfer between twin vertices in a weighted graph with respect to its adjacency, Laplacian and signless Laplacian matrices. As an application, we provide characterizations of all simple unweighted double cones on regular graphs that exhibit periodicity, perfect state transfer, and pretty good state transfer.

math.CO↗

arXiv2021

State Transfer on Paths with Weighted Loops

We consider the fidelity of state transfer on an unweighted path on $n$ vertices, where a loop of weight $w$ has been appended at each of the end vertices. It is known that if $w$ is transcendental, then there is pretty good state transfer from one end vertex to the other; we prove a companion result to that fact, namely that there is a dense subset of $[1,\infty)$ such that if $w$ is in that subset, pretty good state transfer between end vertices is impossible. Under mild hypotheses on $w$ and $t$, we derive upper and lower bounds on the fidelity of state transfer between end vertices at readout time $t$. Using those bounds, we localise the readout times for which that fidelity is close to $1$. We also provide expressions for, and bounds on, the sensitivity of the fidelity of state transfer between end vertices, where the sensitivity is with respect to either the readout time or the weight $w$. Throughout, the results rely on detailed knowledge of the eigenvalues and eigenvectors of the associated adjacency matrix.

quant-ph↗

arXiv2021

Stochastic Matrices Realising the Boundary of the Karpelevi\v c Region

A celebrated result of Karpelevi\v c describes $Θ_n,$ the collection of all eigenvalues arising from the stochastic matrices of order $n.$ The boundary of $Θ_n$ consists of roots of certain one-parameter families of polynomials, and those polynomials are naturally associated with the so--called reduced Ito polynomials of Types 0, I, II and III. In this paper we explicitly characterise all $n \times n$ stochastic matrices whose characteristic polynomials are of Type 0 or Type I, and all sparsest stochastic matrices of order $n$ whose characteristic polynomials are of Type II or Type III. The results provide insights into the structure of stochastic matrices having extreme eigenvalues.

math.SP↗

arXiv2020

The Karpelevič Region Revisited

We consider the Karpelevič region $Θ_n \subset \mathbb{C}$ consisting of all eigenvalues of all stochastic matrices of order $n$. We provide an alternative characterisation of $Θ_n$ that sharpens the original description given by Karpelevič. In particular, for each $θ\in [0, 2π),$ we identify the point on the boundary of $Θ_n$ with argument $θ.$ We further prove that if $n \in \mathbb{N}$ with $n \ge 2,$ and $t \in Θ_n,$ then $t$ is a subdominant eigenvalue of some stochastic matrix of order $n$.

math.SP↗

arXiv2013

alpha-Kuramoto partitions: graph partitions from the frustrated Kuramoto model generalise equitable partitions

The Kuramoto model describes the collective dynamics of a system of coupled oscillators. An alpha-Kuramoto partition is a graph partition induced by the Kuramoto model, when the oscillators include a phase frustration parameter. We prove that every equitable partition is an alpha-Kuramoto partition, but that the converse does necessarily not hold. We give an exact characterisation of alpha-Kuramoto bipartitions.

math.CO↗

arXiv2012

Number-Theoretic Nature of Communication in Quantum Spin Systems

The last decade has witnessed substantial interest in protocols for transferring information on networks of quantum mechanical objects. A variety of control methods and network topologies have been proposed, on the basis that transfer with perfect fidelity --- i.e. deterministic and without information loss --- is impossible through unmodulated spin chains with more than a few particles. Solving the original problem formulated by Bose [Phys. Rev. Lett. 91, 207901 (2003)], we determine the exact number of qubits in unmodulated chains (with XY Hamiltonian) that permit the transfer with fidelity arbitrarily close to 1, a phenomenon called pretty good state transfer. We prove that this happens if and only if the number of nodes is n=p-1, 2p-1, where p is a prime, or n=2^{m}-1. The result highlights the potential of quantum spin system dynamics for reinterpreting questions about the arithmetic structure of integers, and, in this case, primality.

quant-ph↗