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bibkey: coutinho2025peak authors: Gabriel Coutinho, Krystal Guo, Vincent Schmeits year: 2025 title: “Peak state transfer in continuous quantum walks” doi: 10.48550/arXiv.2505.11986 url: https://arxiv.org/abs/2505.11986v4 claim: “Section 3 defines the bounding matrix and peak state transfer; Lemma 5.2 characterizes phase alignment; Open Problem 7.1 asks for infinitely many non-star trees admitting Laplacian peak state transfer.” strata_touched:

• D5/S3/Quantum/Dynamics/LaplacianPeakTransferTrees license: citation-only triage: anchor

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DOI: 10.48550/arXiv.2505.11986

Source: https://arxiv.org/abs/2505.11986v4

DOI URL: https://doi.org/10.48550/arXiv.2505.11986

Definitions and phase alignment

Section 3, p. 5, uses the distinct-eigenvalue spectral decomposition $M={\sum }_{r=0}^d{\theta }_r{E}_r$ and states:

$E_r$ is the idempotent projection onto the ${\theta }_r$-eigenspace.

In particular, the transition matrix of the continuous-time quantum walk on $M$ can be written as the following matrix-valued function in time: $U(t)=e^{itM}={\sum }_{r=0}^d{e}^{it{\theta }_r}{E}_r$.

We will refer to $B(M):={\sum }_{r=0}^d|(E_r)|$ 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.

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 $\tau$ such that $|U(\tau {)}_{v,u}|=B(M{)}_{v,u}$.

Laplacian dynamics uses $M=L(G)=D-A$. Lemma 5.2, p. 8, states that the triangle bound is attained exactly when the phases on the positive and negative mutual spectral support are respectively $\gamma$ and $-\gamma$ for a common complex phase.

Open Problem 7.1

Section 7, p. 21, states:

Determine whether infinitely many such trees, not isomorphic to the star graph, admit Laplacian peak state transfer.

Figure 10, p. 22, depicts $K_{1,3}$, $K_{1,4}$, and a 10-vertex rooted tree with three hubs and two leaves on each hub. The source presents these as examples found by computation.

The Lean definitions use G.lapMatrix ℝ, the eigenbasis-filtered sum of outer products for each spectral idempotent, a sum over the finite image of eigenvalues for boundingEntry, and the complex matrix exponential for lapPropagator. The new result constructs arbitrarily large non-star trees and proves peak transfer from the root to each hub at time $\pi$.