arXiv subjects

Hermie Monterde

Publications and source records attributed to Hermie Monterde.

At least 19 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↗

arXiv2026

Laplacian state transfer in graphs with involutions

For $q\in\mathbb{R}\backslash\{0\}$, the generalized Laplacian of a graph $X$ is the matrix $\mathscr{L}=\Delta+qA$, where $\Delta$ is the degree matrix and $A$ is the adjacency matrix of $X$. In this paper, we investigate perfect state transfer (PST) on graphs with possible loops equipped with non-trivial involutions, where we take the generalized Laplacian matrix as the Hamiltonian of the underlying spin network. We establish an equivalence between the existence of PST between certain pair (or plus states) in such a graph and PST between vertices in a subgraph induced by the involution. This allows us to prove that for almost all simple unweighted planar graphs (resp., almost all simple unweighted trees), the assignment of loops of weight one to exactly two vertices in the graph produces PST between pair states relative to $\mathscr{L}$. We also show that a path on $n$ vertices admits PST between end vertices relative to $\mathscr{L}$ if and only if $n =2$, or $(n,q)=(3,\frac{k^2-l^2}{8l^2})$ where $k>l$ are integers with $k \not\equiv l \pmod{2}$. For cycles, we show that the addition of an extra edge does not yield PST between vertices relative to Laplacian and signless Laplacian matrices. Furthermore, we show that the addition of a few suitable edges (including loops) in complete bipartite graphs, cycles, and paths yields PST between pair states.

math.CO↗

arXiv2026

Maximum spectral sum of graphs

For a graph $G$ of order $n$, the spectral sum of $G$ is defined to be the sum $\lambda_1(G) + \lambda_2(G)$, where $\lambda_1(G)$ (resp. $\lambda_2(G)$) is the largest (resp. second largest) adjacency eigenvalue of $G$. Ebrahimi, Mohar, Nikiforov and Ahmady (2008) conjectured that the spectral sum \[ \lambda_1(G) + \lambda_2(G)\le \frac{8}{7}n \] for any graph $G$. We prove this conjecture by combining tools from the theory of graph limits, convex geometry, exterior algebra and convex optimization. The techniques developed are of independent interest.

math.CO↗

arXiv2026

Local $\epsilon$-uniform mixing in continuous quantum walks

Let $X$ be a weighted graph and $M$ be its adjacency, Laplacian or signless Laplacian matrix. In a continuous quantum walk on $X$, local $\epsilon$-uniform mixing occurs at vertex $u$ if the $u$th column of the matrix $U(t)=e^{itM}$ can be made arbitrarily close to a vector whose all entries have equal magnitude. Using the spectral and combinatorial properties of $X$, we derive necessary conditions for local $\epsilon$-uniform mixing to occur in $X$. This includes an inequality involving all entries of each eigenvector of $M$, as well as an upper bound on the degree of vertex $u$ when $M$ is the Laplacian or signless Laplacian matrix. We use these necessary conditions to rule out local $\epsilon$-uniform mixing in numerous classes of graphs, most of which are non-regular. We also show that almost all planar graphs (resp., trees) contain a vertex that does not admit local $\epsilon$-uniform mixing for any assignment of edge weights. Furthermore, we prove if $X$ has $n$ vertices and admits local $\epsilon$-uniform mixing at a vertex contained in a subgraph with a twin, then the number of vertices of this twin subgraph must be at least $\sqrt{n}$. In particular, we establish that a graph on $n\geq 5$ vertices does not admit local $\epsilon$-uniform mixing at a vertex with a twin.

math.CO↗

arXiv2026

Sedentary quantum walks on bipartite graphs

If a quantum walk starting on a vertex tends to stay at home, then that vertex is said to be sedentary. We prove that almost all planar graphs and almost all trees contain at least two sedentary vertices for any assignment of edge weights -- a result that suggests vertex sedentariness is a common phenomenon in trees and planar graphs. For weighted bipartite graphs, we show that a vertex is not sedentary whenever 0 does not belong to its eigenvalue support. Consequently, each vertex in a nonsingular weighted bipartite graph is not sedentary, a stark contrast to weighted trees and weighted planar graphs. A corollary of this result is that every vertex in a bipartite graph with a unique perfect matching is not sedentary for any assignment of edge weights. We also construct new families of weighted bipartite graphs with sedentary vertices using the bipartite double and subdivision operations. Finally, we show that unweighted paths and unweighted even cycles contain no sedentary vertices.

math.CO↗

arXiv2026

Structured eigenbases and pair state transfer on threshold graphs

Recently, Macharete, Del-Vecchio, Teixeira and de Lima showed that a star and any threshold graph on the same number of vertices share the same eigenbasis relative to the Laplacian matrix. We use this fact to establish two main results in this paper. The first one is a characterization of threshold graphs that are \textit{simply structured}, i.e., their associated Laplacian matrices have eigenbases consisting of vectors with entries from the set $\{-1,0,1\}$. Then, we provide sufficient conditions such that a simply structured threshold graph is weakly Hadamard diagonalizable (WHD). This allows us to list all connected simply structured threshold graphs on at most 20 vertices, and identify those that are WHD. Second, we characterize Laplacian pair state transfer on threshold graphs. In particular, we show that the existence of Laplacian vertex state transfer and Laplacian pair state transfer on a threshold graph are equivalent if and only if it is not a join of a complete graph and an empty graph of certain sizes.

math.CO↗

arXiv2025

Quantum walks on finite and bounded infinite graphs

A weighted graph $G$ with countable vertex set is bounded if there is an upper bound on the maximum of the sum of absolute values of all edge weights incident to a vertex in $G$. In this paper, we prove a fundamental result on equitable partitions of bounded weighted graphs with twin subgraphs and use this fact to construct finite and bounded infinite graphs with pair and plus state transfer with the adjacency matrix as a Hamiltonian. We show that for each $k \ge 3$, (i) there are infinitely many connected unweighted graphs with maximum degree $k$ admitting pair state transfer at $\tau\in\{\frac{\pi}{\sqrt{2}},\frac{\pi}{2}\}$, and (ii) there are infinitely many signed graphs with exactly one negative edge weight and whose underlying unweighted graphs have maximum degree $k$ admitting plus state transfer at $\tau\in\{\frac{\pi}{\sqrt{2}},\frac{\pi}{2}\}$. Parallel results are proven for perfect state transfer between a plus state and a pair state, and for the existence of sedentary pair and plus states. We further prove that almost all connected unweighted finite planar graphs admit pair state transfer at $\tau\in\{\frac{\pi}{\sqrt{2}},\frac{\pi}{2}\}$, and almost all connected unweighted finite planar graphs can be assigned a single negative edge weight resulting in plus state transfer, or perfect state transfer between a plus state and a pair state, at $\tau\in\{\frac{\pi}{\sqrt{2}},\frac{\pi}{2}\}$. Analogous results are shown to hold for unweighted finite trees. Using blow-up graphs, Cayley graphs and graphs with tails, we construct new infinite families of (finite and infinite) unweighted graphs and signed graphs admitting pair or plus state transfer.

math.CO↗

arXiv2025

More graphs with pair state transfer

This paper has two main goals. First, we characterize perfect state transfer between $s$-pair states in strongly regular graphs, as well as graphs in association schemes admitting perfect state transfer between vertices. The second goal is to provide a unified approach for constructing non-regular graphs admitting pair state transfer$-$relative to the adjacency, Laplacian, and signless Laplacian matrix$-$between the same pair of states at the same time. In particular, we show that for each $k\geq 5$, there are infinitely many connected graphs with maximum valency $k$ admitting this property. We also utilize graph products to generate new infinite families of graphs with pair state transfer.

math.CO↗

arXiv2025

Laplacian quantum walks on blow-up graphs

This paper is a sequel to the work of Bhattacharjya et al.\ (J. Phys. A-Math. 57.33: 335303, https://doi.org/10.1088/1751-8121/ad6653) on quantum state transfer on blow-up graphs, where instead of the adjacency matrix, we take the Laplacian matrix as the time-independent Hamiltonian associated with a blow-up graph. We characterize strong cospectrality, periodicity, perfect state transfer (LPST) and pretty good state transfer (LPGST) on blow-up graphs. We present several constructions of blow-up graphs with LPST and produce new infinite families of regular graphs where each vertex is involved in LPST. We also determine LPST and LPGST in blow-ups of classes of trees. Finally, if $n\equiv 0$ (mod 4), then the blow-up of $n$ copies of a graph $G$ has no LPST, but we show that under certain conditions, the addition of an appropriate matching this blow-up graph results in LPST.

math.CO↗

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

Weakly Hadamard diagonalizable graphs and Quantum State Transfer

Hadamard diagonalizable graphs are undirected graphs for which the corresponding Laplacian is diagonalizable by a Hadamard matrix. Such graphs have been studied in the context of quantum state transfer. Recently, the concept of a weak Hadamard matrix was introduced: a $\{-1,0, 1\}$-matrix $P$ such that $PP^T$ is tridiagonal, as well as the concept of weakly Hadamard diagonalizable graphs. We therefore naturally explore quantum state transfer in these generalized Hadamards. Given the infancy of the topic, we provide numerous properties and constructions of weak Hadamard matrices and weakly Hadamard diagonalizable graphs in order to better understand them.

math.CO↗

arXiv2024

Quantum walks on blow-up graphs

A blow-up of $n$ copies of a graph $G$ is the graph $\overset{n}\uplus~G$ obtained by replacing every vertex of $G$ by an independent set of size $n$, where the copies of vertices in $G$ are adjacent in the blow-up if and only if the vertices adjacent in $G$. Our goal is to investigate the existence of quantum state transfer on a blow-up graph $\overset{n}\uplus~G$, where the adjacency matrix is taken to be the time-independent Hamiltonian of the quantum system represented by $\overset{n}\uplus~G$. In particular, we establish necessary and sufficient conditions for vertices in a blow-up graph to exhibit strong cospectrality and various types of high probability quantum transport, such as periodicity, perfect state transfer (PST) and pretty good state transfer (PGST). It turns out, if $\overset{n}\uplus~G$ admits PST or PGST, then one must have $n=2.$ Moreover, if $G$ has an invertible adjacency matrix, then we show that every vertex in $\overset{2}\uplus~G$ pairs up with a unique vertex to exhibit strong cospectrality. We then apply our results to determine infinite families of graphs whose blow-ups admit PST and PGST.

quant-ph↗

arXiv2023

New results in vertex sedentariness

A vertex in a graph is said to be sedentary if a quantum state assigned on that vertex tends to stay on that vertex. Under mild conditions, we show that the direct product and join operations preserve vertex sedentariness. We also completely characterize sedentariness in blow-up graphs. These results allow us to construct new infinite families of graphs with sedentary vertices. We prove that a vertex with a twin is either sedentary or admits pretty good state transfer. Moreover, we give a complete characterization of twin vertices that are sedentary, and provide sharp bounds on their sedentariness. As an application, we determine the conditions in which perfect state transfer, pretty good state transfer and sedentariness occur in complete bipartite graphs and threshold graphs of any order.

math.CO↗

arXiv2023

Quantum walks on join graphs

The join $X\vee Y$ of two graphs $X$ and $Y$ is the graph obtained by joining each vertex of $X$ to each vertex of $Y$. We explore the behaviour of a continuous quantum walk on a weighted join graph having the adjacency matrix or Laplacian matrix as its associated Hamiltonian. We characterize strong cospectrality, periodicity and perfect state transfer (PST) in a join graph. We also determine conditions in which strong cospectrality, periodicity and PST are preserved in the join. Under certain conditions, we show that there are graphs with no PST that exhibits PST when joined by another graph. This suggests that the join operation is promising in producing new graphs with PST. Moreover, for a periodic vertex in $X$ and $X\vee Y$, we give an expression that relates its minimum periods in $X$ and $X\vee Y$. While the join operation need not preserve periodicity and PST, we show that $\big| |U_M(X\vee Y,t)_{u,v}|-|U_M(X,t)_{u,v}| \big|\leq \frac{2}{|V(X)|}$ for all vertices $u$ and $v$ of $X$, where $U_M(X\vee Y,t)$ and $U_M(X,t)$ denote the transition matrices of $X\vee Y$ and $X$ respectively relative to either the adjacency or Laplacian matrix. We demonstrate that the bound $\frac{2}{|V(X)|}$ is tight for infinite families of graphs.

quant-ph↗

arXiv2023

Sedentariness in quantum walks

We formalize the notion of a sedentary vertex and present a relaxation of the concept of a sedentary family of graphs introduced by Godsil [Linear Algebra Appl. 614:356-375, 2021]. We provide sufficient conditions for a given vertex in a graph to exhibit sedentariness. We also show that a vertex with at least two twins (vertices that share the same neighbours) is sedentary. We prove that there are infinitely many graphs containing strongly cospectral vertices that are sedentary, which reveals that, even though strong cospectrality is a necessary condition for pretty good state transfer, there are strongly cospectral vertices which resist high probability state transfer to other vertices. Moreover, we derive results about sedentariness in products of graphs which allow us to construct new sedentary families, such as Cartesian powers of complete graphs and stars.

quant-ph↗

arXiv2023

Fractional revival between twin vertices

In this paper, we provide a characterization of fractional revival between twin vertices in a weighted graph with respect to its adjacency, Laplacian and signless Laplacian matrices. As an application, we characterize fractional revival between apexes of double cones.

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↗