Infinitely many non-star trees admit Laplacian peak state transfer

Abstract

There are arbitrarily large non-star trees with Laplacian peak state transfer. For each positive odd s, join a root to s(s+1)+1 hubs and each hub to s(s+1) leaves. Transfer from the root to every hub attains the spectral entry bound at time pi.

Lemma 1.1 (Distinct eigenvalues give orthogonal vectors).

$$\forall (V:\mathrm{Type}),[\mathrm{Fintype}\left(V\right)],[\mathrm{DecidableEq}\left(V\right)],\forall (A:\mathrm{Matrix}\left(V,V,\mathbb{R}\right)),\forall (hA:\mathrm{Matrix}.\mathrm{IsHermitian}\left(A\right)),\forall (w:V\Rightarrow \mathbb{R}),\forall (mu:\mathbb{R}),\mathrm{Matrix}.\mathrm{mulVec}\left(A,w\right)=\mathrm{HSMul}.\mathrm{hSMul}\left(mu,w\right)\Rightarrow \forall (i:V),hA.\mathrm{eigenvalues}\left(i\right)\ne mu\Rightarrow \mathrm{dotProduct}\left(\mathrm{WithLp}.\mathrm{ofLp}\left(hA.\mathrm{eigenvectorBasis}\left(i\right)\right),w\right)=0$$

Proof. Machine-checked in Lean as D5/S3/Quantum/Dynamics/LaplacianPeakTransferTrees.eig_dot_zero (✓ std3). ∎

Source. Repository-derived.

Acknowledgement. Gabriel Coutinho, Krystal Guo, Vincent Schmeits (2025). Peak state transfer in continuous quantum walks. DOI: 10.48550/arXiv.2505.11986. URL: https://arxiv.org/abs/2505.11986v4.

Commentary.

Let A be a real Hermitian matrix on a finite index type V, and let w satisfy A w = mu w. If the eigenvalue of the i-th vector in hA.eigenvectorBasis differs from mu, its dot product with w is zero. The conclusion includes the zero vector and does not require a graph Laplacian. It follows from the orthogonality of distinct eigenspaces of the symmetric Euclidean linear map defined by A.

Definition 1.2 (Laplacian spectral idempotents).

$$\forall (n:\mathbb{N}),\forall (G:\mathrm{SimpleGraph}\left(\mathrm{Fin}\left(n\right)\right)),[\mathrm{DecidableRel}\left(G.\mathrm{Adj}\right)],\forall (theta:\mathbb{R}),\mathrm{lapIdempotent}\left(G,theta\right)=\mathrm{let}hL=G.\mathrm{isHermitia}{\mathrm{n}}_{\mathrm{lapMatrix}}\left(\mathbb{R}\right);\sum _{i:\mathrm{Fin}\left(n\right),hL.\mathrm{eigenvalues}\left(i\right)=theta}\mathrm{vecMulVec}\left(\mathrm{WithLp}.\mathrm{ofLp}\left(hL.\mathrm{eigenvectorBasis}\left(i\right)\right),\mathrm{WithLp}.\mathrm{ofLp}\left(hL.\mathrm{eigenvectorBasis}\left(i\right)\right)\right)$$

Formalization. D5/S3/Quantum/Dynamics/LaplacianPeakTransferTrees.lapIdempotent (✓ std3).

Citation. Gabriel Coutinho, Krystal Guo, Vincent Schmeits (2025). Peak state transfer in continuous quantum walks. DOI: 10.48550/arXiv.2505.11986. URL: https://arxiv.org/abs/2505.11986v4.

Commentary.

Section 3, p. 5: “E_r is the idempotent projection onto the θ_r-eigenspace.” The source writes M as the sum of θ_r E_r over its distinct eigenvalues. Here M is G.lapMatrix over the reals. The formula sums the outer products of the real orthonormal eigenbasis vectors whose eigenvalue equals theta; a missing eigenvalue gives the zero matrix. hL denotes G.isHermitian_lapMatrix over the reals.

Definition 1.3 (An entry of the bounding matrix).

$$\forall (n:\mathbb{N}),\forall (G:\mathrm{SimpleGraph}\left(\mathrm{Fin}\left(n\right)\right)),[\mathrm{DecidableRel}\left(G.\mathrm{Adj}\right)],\forall (v:\mathrm{Fin}\left(n\right)),\forall (u:\mathrm{Fin}\left(n\right)),\mathrm{boundingEntry}\left(G,v,u\right)=\mathrm{let}hL=G.\mathrm{isHermitia}{\mathrm{n}}_{\mathrm{lapMatrix}}\left(\mathbb{R}\right);\sum _{theta\in \mathrm{Finset}.\mathrm{image}\left(hL.\mathrm{eigenvalues},\mathrm{univ}\right)}\left|\mathrm{lapIdempotent}\left(G,theta,v,u\right)\right|$$

Formalization. D5/S3/Quantum/Dynamics/LaplacianPeakTransferTrees.boundingEntry (✓ std3).

Citation. Gabriel Coutinho, Krystal Guo, Vincent Schmeits (2025). Peak state transfer in continuous quantum walks. DOI: 10.48550/arXiv.2505.11986. URL: https://arxiv.org/abs/2505.11986v4.

Commentary.

Section 3, p. 5: “We will refer to B(M) := ∑{r=0}^d |(E_r)| as the bounding matrix of M, as the (v,u)-entry of B(M) upper-bounds |U(t){v,u}| for all values of t.” Absolute values are entrywise. The finite image of hL.eigenvalues counts each distinct eigenvalue once. The two arguments v,u specify the (v,u)-entry, and hL is G.isHermitian_lapMatrix over the reals.

Definition 1.4 (The Laplacian propagator).

$$\forall (n:\mathbb{N}),\forall (G:\mathrm{SimpleGraph}\left(\mathrm{Fin}\left(n\right)\right)),[\mathrm{DecidableRel}\left(G.\mathrm{Adj}\right)],\forall (tau:\mathbb{R}),\mathrm{lapPropagator}\left(G,tau\right)=\mathrm{ProjectionProbabilityFlow}.\mathrm{hamiltonianPropagator}\left(\mathrm{Matrix}.\mathrm{map}\left(G.\mathrm{lapMatrix}\left(\mathbb{R}\right),\mathrm{algebraMap}\left(\mathbb{R},\mathbb{C}\right)\right),-tau\right)$$

Formalization. D5/S3/Quantum/Dynamics/LaplacianPeakTransferTrees.lapPropagator (✓ std3).

Citation. Gabriel Coutinho, Krystal Guo, Vincent Schmeits (2025). Peak state transfer in continuous quantum walks. DOI: 10.48550/arXiv.2505.11986. URL: https://arxiv.org/abs/2505.11986v4.

Commentary.

Section 3, p. 5: “In particular, the transition matrix of the continuous-time quantum walk on M can be written as the following matrix-valued function in time:” U(t) = e^{itM} = ∑_{r=0}^d e^{itθ_r} E_r. For Laplacian dynamics M is G.lapMatrix over the reals, mapped entrywise into the complex numbers by algebraMap. ProjectionProbabilityFlow.hamiltonianPropagator at time -tau equals exp(i tau L), the source’s propagator.

Definition 1.5 (Peak state transfer).

$$\forall (n:\mathbb{N}),\forall (G:\mathrm{SimpleGraph}\left(\mathrm{Fin}\left(n\right)\right)),[\mathrm{DecidableRel}\left(G.\mathrm{Adj}\right)],\forall (u:\mathrm{Fin}\left(n\right)),\forall (v:\mathrm{Fin}\left(n\right)),\mathrm{PeakTransfer}\left(G,u,v\right)\iff ((u\ne v)\wedge (\exists (tau:\mathbb{R}),\Vert \mathrm{lapPropagator}\left(G,tau,v,u\right)\Vert =\mathrm{boundingEntry}\left(G,v,u\right)))$$

Formalization. D5/S3/Quantum/Dynamics/LaplacianPeakTransferTrees.PeakTransfer (✓ std3).

Citation. Gabriel Coutinho, Krystal Guo, Vincent Schmeits (2025). Peak state transfer in continuous quantum walks. DOI: 10.48550/arXiv.2505.11986. URL: https://arxiv.org/abs/2505.11986v4.

Commentary.

Section 3, p. 5: “For distinct vertices u,v, we say that there is peak state transfer from u to v with respect to M if there exists a time τ such that |U(τ){v,u}| = B(M){v,u}.” Here tau encodes τ, M is the real Laplacian, and the complex norm is its absolute value. The equality is at the (v,u)-entry, so u is the input vertex.

Definition 1.6 (Open Problem 7.1).

$$claim\iff (\forall (N:\mathbb{N}),\exists (n:\mathbb{N}),(n\ge N)\wedge (\exists (T:\mathrm{SimpleGraph}\left(\mathrm{Fin}\left(n\right)\right)),(T.\mathrm{IsTree})\wedge ((\forall (m:\mathbb{N}),\mathrm{IsEmpty}\left(\mathrm{SimpleGraph}.\mathrm{Iso}\left(T,\mathrm{completeBipartiteGraph}\left(\mathrm{Fin}\left(1\right),\mathrm{Fin}\left(m\right)\right)\right)\right))\wedge (\exists (u:\mathrm{Fin}\left(n\right)),\exists (v:\mathrm{Fin}\left(n\right)),\mathrm{PeakTransfer}\left(T,u,v\right)))))$$

Formalization. D5/S3/Quantum/Dynamics/LaplacianPeakTransferTrees.claim (✓ std3).

Citation. Gabriel Coutinho, Krystal Guo, Vincent Schmeits (2025). Peak state transfer in continuous quantum walks. DOI: 10.48550/arXiv.2505.11986. URL: https://arxiv.org/abs/2505.11986v4.

Commentary.

Section 7, p. 21: “Determine whether infinitely many such trees, not isomorphic to the star graph, admit Laplacian peak state transfer.” Figure 10, p. 22, shows two stars and the 10-vertex rooted tree with three hubs and two leaves per hub. In the formula N ranges over natural lower bounds on the number n of vertices, T is a simple graph on Fin n, and IsTree means connected and acyclic. The exclusion covers every star K_(1,m), including m=0, through the absence of a SimpleGraph.Iso to completeBipartiteGraph (Fin 1) (Fin m). The PeakTransfer instances use classical adjacency decidability. Arbitrarily large finite orders imply infinitely many isomorphism classes.

Theorem 1.7 (The answer is affirmative).

$$claim$$

Proof. Machine-checked in Lean as D5/S3/Quantum/Dynamics/LaplacianPeakTransferTrees.result (✓ std3). ∎

Resolves. Problems/coutinho-guo-schmeits-2025-laplacian-peak-transfer-trees (proved) by D5/S3/Quantum/Dynamics/LaplacianPeakTransferTrees.result.

Source. Repository-derived.

Acknowledgement. Gabriel Coutinho, Krystal Guo, Vincent Schmeits (2025). Peak state transfer in continuous quantum walks. DOI: 10.48550/arXiv.2505.11986. URL: https://arxiv.org/abs/2505.11986v4.

Commentary.

Put l=s(s+1), k=l+1 and n=1+k(l+1), with s positive and odd. The root, hubs and leaves give a subspace of cell-constant vectors preserved by the Laplacian. The root basis vector decomposes into eigenvectors with eigenvalues 0, s^2+1 and (s+1)^2+1. Their root-to-hub spectral entries have signs positive, positive and negative. At time pi the corresponding phases are 1, 1 and -1, so the absolute value of the propagator entry equals the sum of the absolute spectral entries. This is the phase-alignment mechanism of Lemma 5.2. Each graph is connected and has n-1 edges. The root and a hub both have degree at least two, which excludes all stars. Taking s=2N+1 yields a graph whose order is at least N. The resulting transfer is across one edge.

References

• Truth anchor: D5/S3/Quantum/Dynamics/LaplacianPeakTransferTrees.PeakTransfer
• Truth anchor: D5/S3/Quantum/Dynamics/LaplacianPeakTransferTrees.boundingEntry
• Truth anchor: D5/S3/Quantum/Dynamics/LaplacianPeakTransferTrees.claim
• Truth anchor: D5/S3/Quantum/Dynamics/LaplacianPeakTransferTrees.eig_dot_zero
• Truth anchor: D5/S3/Quantum/Dynamics/LaplacianPeakTransferTrees.lapIdempotent
• Truth anchor: D5/S3/Quantum/Dynamics/LaplacianPeakTransferTrees.lapPropagator
• Truth anchor: D5/S3/Quantum/Dynamics/LaplacianPeakTransferTrees.result
• Dependency: D5/S3/Quantum/Dynamics/EnergyEigenstateStationarity
• Dependency: D5/S3/Quantum/Dynamics/ProjectionProbabilityFlow
• Dependency: D5/S3/Quantum/Dynamics/RationalWeightPathTransfer