bibkey: kay2010perfectstatetransferreview authors: Alastair Kay year: 2010 title: “A Review of Perfect State Transfer and its Application as a Constructive Tool” doi: 10.1142/S0219749910006514 url: https://arxiv.org/abs/0903.4274v3 claim: “Lemma 6 (subsection Transfer Rate): for eigenvalues fulfilling the perfect state transfer condition, the rate M/2t_0 is perfectly achieved for an integer M if and only if the sums R_k of (-1)^n/B’(lambda_n) over the residue classes of (t_0/pi)(lambda_n - lambda_1) modulo M, k = 0, …, M - 1, are all equal. The review conjectures that this condition cannot be fulfilled for any M > 2 and states that it has a proof for M > N/2.” strata_touched:

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

A review of perfect state transfer and its application as a constructive tool

A. Kay, A Review of Perfect State Transfer and its Application as a Constructive Tool, arXiv:0903.4274v3 (16 June 2010); Int. J. Quantum Inf. 8, 641 (2010). The quotations are taken from the arXiv v3 source review.tex.

The perfect state transfer condition

The subsection “The Symmetry Matching Condition” orders the eigenvalues of a mirror-symmetric nearest-neighbour chain and states the condition, Eqn. eqn:st_cond:

Consider $\{{\lambda }_n\}$ to be an ordered set of eigenvalues, ${\lambda }_n<{\lambda }_{n+1}$. The necessary and sufficient conditions for state transfer in a symmetric chain then become that \begin{equation} \lambda_n-\lambda_{n-1}=(2m_n+1)\pi/t_0, \label{eqn:st_cond} \end{equation} where $t_0$ is the state transfer time, and $m_n$ is a positive integer (which can vary with $n$).

The subsection “Inverse Eigenvalue Problems” introduces

$$>B^{\prime }({\lambda }_n)=\prod _{m=1\ne n}^N({\lambda }_n-{\lambda }_m),>$$ which is the derivative of the function $B(\lambda )={\prod }_{m=1}^N(\lambda -{\lambda }_m)$, the characteristic polynomial of the system, evaluated at ${\lambda }_n$.

and recalls the weights of the eigenvectors on the first site, Eqn. eqn:hoch:

In \cite{hochstadt}, it is proven that for a symmetric tridiagonal matrix, \begin{equation} |\alpha_n|^2=\frac{1}{(-1)^nB’(\lambda_n)\sum_{m=1}^N\frac{1}{(-1)^nB’(\lambda_m)}}. \label{eqn:hoch} \end{equation}

The rate lemma and the conjecture

The subsection “Transfer Rate” asks at which rate $1/t_r$ single-qubit states can be sent through the chain when a new state is placed on the first spin after the time $t_r$, and finds that the amplitude ${\gamma }_1$ on the first site must vanish at that time. It continues:

The following lemma allows us to prove a tighter bound of $1/t_r\le N/4t_0$, although we conjecture a stronger condition; that there are no chains with $t_r<t_0$.

The lemma (lemma:rate, the sixth lemma of the source) reads:

\begin{lemma} If a set of eigenvalues is chosen to fulfill the perfect state transfer condition of Eqn.~(\ref{eqn:st_cond}), then a necessary and sufficient condition to perfectly achieve the rate $M/2t_0$ for integer $M$ is that all the $R_k$ for $k=0\dots M-1$ should be equal, where \begin{equation} R_k=\sum_{n=1}^N\frac{(-1)^n}{B’(\lambda_n)}, \label{eqn:rate_cond} \end{equation} and the sum is restricted to those terms satisfying the condition $$>\frac{t_0}{\pi }({\lambda }_n-{\lambda }_1)\mathrm{mod}M=k.>$$ \label{lemma:rate} \end{lemma}

Its proof writes ${\gamma }_1(t)$ through Eqn. eqn:hoch and demands ${\gamma }_1(2m{t}_0/M)=0$ for $m=1\dots M-1$ and ${\gamma }_1(2t_0)=1$. After a remark on composite $M$ the source states the conjecture:

We conjecture that it is impossible to fulfill the condition of Lemma \ref{lemma:rate} for any $M>2$, although we only have a proof for $M>N/2$.

The module D5/S3/Quantum/Dynamics/KayTransferRateRefutation refutes this conjecture: the eight eigenvalues $0,31,46,65,88,107,122,153$ with $t_0=\pi$ fulfil Eqn. eqn:st_cond with $m=15,7,9,11,9,7,15$, and for $M=4=N/2$ all four sums $R_k$ equal $194/38984495395755$.

Verified locator

• DOI: https://doi.org/10.1142/S0219749910006514 (the DOI 10.1142/S0219749910006514 and the journal reference Int. J. Quantum Inf. 8, 641 (2010) are those listed by the arXiv API for 0903.4274, read 2026-10-08).
• arXiv: https://arxiv.org/abs/0903.4274v3 (source file review.tex): Eqn. eqn:st_cond in the subsection “The Symmetry Matching Condition”, $B^{\prime }$ and Eqn. eqn:hoch in the subsection “Inverse Eigenvalue Problems”, the lemma lemma:rate and the conjecture in the subsection “Transfer Rate”, the fourth subsection of the third section, “Higher Excitation Subspaces”.