bibkey: escobar2025earlystateexclusion authors: Mia Gabriella Escobar; Valentin Garcia; Anastasiia Minenkova year: 2025 title: “Early State Exclusion in 7-Qubit Spin Chains” doi: 10.48550/arXiv.2507.18767 url: https://arxiv.org/abs/2507.18767v1 claim: “Section 4 conjectures that a 7 x 7 Jacobi matrix with symmetric spectrum realizing perfect state transfer does not have early state exclusion if and only if its positive eigenvalues are integer multiples of the smallest positive eigenvalue; Theorems 3.2 and 3.4 prove the spectra {0, +-1, +-2m, +-(2m+1)} (no early state exclusion) and {0, +-(2m+1), +-(2m+2), +-(2m+3)} (early state exclusion exactly 2m times).” strata_touched:

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Early state exclusion in 7-qubit spin chains

M. G. Escobar, V. Garcia and A. Minenkova, Early State Exclusion in 7-Qubit Spin Chains, arXiv:2507.18767v1 (24 July 2025). The arXiv record lists these three authors; the title page of the source names M. G. Escobar and V. Garcia as authors and A. Minenkova as project advisor. The quotations are taken from the arXiv v1 source Finaldraft7x7.tex.

Jacobi matrices, perfect state transfer and early state exclusion

The introduction fixes the Hamiltonian of a chain of $N+1$ qubits as the Jacobi matrix of Eq. (1.1), with diagonal entries $a_1,\dots ,a_{N+1}$ and off-diagonal entries $b_1,\dots ,b_N$,

for $a_i\in \mathbb{R}$ and $b_i>0$.

Perfect state transfer is defined by:

A Jacobi matrix $J$ realizes \textbf{perfect state transfer} (PST) between the end-vertices of the weighted path in Figure \ref{fig:nearestneighbor} at time $T>0$ if \begin{equation} e^{-iJT}\e_0 = e^{i\phi}\e_N \end{equation} for some phase $\varphi \in \mathbb{R}$.

The source recalls Kay’s criterion (persymmetry together with ${\mu }_{k+1}-{\mu }_k=\frac{(2n_k+1)\pi }{T}$ for the ordered eigenvalues) and restricts to symmetric spectra:

We consider a special case where $J$ has symmetric spectrum in section 3, meaning that for $\{{\lambda }_k{\}}_{k=0}^N$, ${\lambda }_j=-{\lambda }_{N-j}$ for all $1\le j\le N-1$. If we assume that the positive eigenvalues of $J$ are coprime, then \eqref{1.3} implies that $J$ first realizes PST at time $T=\pi$.

Early state exclusion is defined by:

Let $J$ be a Jacobi matrix realizing PST at earliest time $T>0$. If there exists some $\tau \in (0,T)$ such that \begin{equation} \langle e^{-iJ\tau}\e_0, \e_0 \rangle = 0, \end{equation} then we say that $J$ exhibits \textbf{early state exclusion} (ESE) at time $\tau$.

The first-site amplitude of a 7-site chain

For the symmetric spectrum $\{0,\pm x,\pm y,\pm z\}$, $0<x<y<z$, the source gives the persymmetric Jacobi matrix (zero diagonal, $b_1=xz/\sqrt{x^2-y^2+z^2}$, $b_2=\sqrt{(y^2-x^2)(z^2-y^2)/(x^2-y^2+z^2)}$, $b_3=\sqrt{(x^2-y^2+z^2)/2}$) and the amplitude, Eq. (2.1):

\begin{align} \langle e^{-iJt}\e_0, \e_0 \rangle &= \frac{1}{2y^2(z^2 - x^2)(x^2-y^2+z^2)}\Big((y^2 - x^2)(z^2 - x^2)(z^2 - y^2)\ &\quad + y^2z^2(z^2 - y^2)\cos{xt} + x^2z^2(z^2 - x^2)\cos{yt} + x^2y^2(y^2 - x^2)\cos {zt}\Big). \end{align}

The proved families and the conjecture

Theorem 3.2:

Suppose that $J$ is a Jacobi matrix of order $7$ realizing PST with symmetric spectrum [ {0, \pm 1, \pm 2m, \pm (2m + 1)} ] for some integer $m\ge 1$. Then $J$ \textbf{does not} have ESE.

Theorem 3.4:

Let $m\ge 1$ be an integer. Then the Jacobi matrix $J$ realizing PST with symmetric spectrum $ {0, \pm (2m + 1), \pm (2m + 2), \pm (2m + 3)}$ has ESE $2m$ times.

Section 4, “Observations and Future Work”, states:

\begin{conjecture} Let $J$ be a $7\times 7$ Jacobi matrix with symmetric spectrum realizing PST. Then $J$ does not have ESE if and only if its positive eigenvalues are integer multiples of the smallest positive eigenvalue. \end{conjecture} We believe that the techniques used to prove Theorems \ref{Theorem 3.2} and \ref{Theorem 3.4} cannot be replicated here.

The module D5/S3/Quantum/Dynamics/EndAmplitudeCosineSumZeros takes the bracket of Eq. (2.1) at natural frequencies $(x,y,z)=(a,b,c)$ as its definition amplitudeNumerator and proves, for odd $a$ < even $b$ < odd $c$ with $\gcd (a,b,c)=1$, that this cosine sum has a zero in $(0,\pi )$ if and only if $a\ne 1$. The triples $(1,2m,2m+1)$ of Theorem 3.2 and $(2m+1,2m+2,2m+3)$ of Theorem 3.4 are special cases; the count of exactly $2m$ zeros in Theorem 3.4 is not treated in that module.

Verified locator

• DOI: https://doi.org/10.48550/arXiv.2507.18767 (the arXiv-issued DOI; the arXiv API record of 2507.18767 carries no journal reference and no publisher DOI, read 2026-10-08).
• arXiv: https://arxiv.org/abs/2507.18767v1 (source file Finaldraft7x7.tex): Eq. (1.1), the definition of perfect state transfer, Eq. (1.3) and the definition of early state exclusion in Section 1; Eq. (2.1) in the subsection “$7\times 7$ Persymmetric Jacobi Matrices with Symmetric Spectrum”; Theorems 3.2 and 3.4 in Section 3; the conjecture in Section 4.