bibkey: kirkland2019rationalweightspst authors: S. Kirkland, D. McLaren, R. Pereira, S. Plosker and X. Zhang year: 2019 title: “Perfect quantum state transfer in weighted paths with potentials (loops) using orthogonal polynomials” doi: 10.1080/03081087.2018.1442810 url: https://arxiv.org/abs/1708.03283v2 claim: “Rational weights conjecture: a weighted path on at least four vertices, with or without potentials, whose edge weights are all rational has no adjacency matrix perfect state transfer between its end vertices at readout time π. Proposition prop:rat proves it for n = 4 and for n ≥ 5 with n ≡ 3 or 5 (mod 8).” strata_touched:

• D5/S3/Quantum/Dynamics/RationalWeightPathTransfer
• D5/S3/Quantum/Dynamics/PathMiddleVertexMoments
• D5/S3/Quantum/Dynamics/OddPathRationalWeightTransfer license: citation-only triage: anchor

Perfect state transfer on weighted paths with rational weights

S. Kirkland, D. McLaren, R. Pereira, S. Plosker and X. Zhang, Perfect quantum state transfer in weighted paths with potentials (loops) using orthogonal polynomials, arXiv:1708.03283v2 (22 March 2019); Linear and Multilinear Algebra 67(5) (2019) 1043–1061.

The Hamiltonian of a weighted path on $n$ vertices is the tridiagonal adjacency matrix with the potentials $q_j$ on the diagonal and the positive edge weights $r_j$ off the diagonal. The source defines perfect state transfer verbatim:

There is \emph{perfect state transfer} from vertex $j$ to vertex $k$ if there exists some time $t=t_0$ such that $|e_j^T{e}^{i{t}_{0}H}{e}_k{|}^2=1$

The introduction states the conjecture:

For XX dynamics, our analysis leads us to propose the following conjecture: weighted paths on at least four vertices with or without loops must have at least one irrational weight in order to have adjacency matrix PST at a fixed readout time $\pi$; we confirm this conjecture for $n=4$ as well as for n\equiv 3\modn 8 and for n\equiv 5\modn 8.

The section “Adjacency Matrices and the Rational weights conjecture” repeats it:

We have a conjecture about the weights: if all the weights of a weighted path on at least 4 vertices are rational numbers, then there is no adjacency matrix PST at time $\pi$ between the end vertices of the path. We confirm that conjecture in the cases that $n=4$, n\equiv 5\modn 8 and for n\equiv 3\modn 8 but $n\ne 3$.

Proposition prop:rat:

Suppose that $n=4,$ or $n\ge 5$ and n\equiv 3\modn 8 or n\equiv 5\modn 8. If the weights of a weighted path on $n$ vertices with or without potentials are all rational numbers, then there is no adjacency matrix PST between its end vertices at readout time $\pi$.

The weights are the edge weights $r_j$; the potentials are unrestricted real numbers.

The module D5/S3/Quantum/Dynamics/RationalWeightPathTransfer proves the statement of the conjecture for every $n=2^k+1$ with $k\ge 1$, in both directions between the end vertices; the source does not single out this family. This is partial progress on the conjecture: $k=1$ gives $n=3$, below the conjecture’s range $n\ge 4$; $k=2$ gives $n=5$, one of the cases of Proposition prop:rat; the new sizes are those with $k\ge 3$, namely $n=9,17,33,\dots$, which satisfy $n\equiv 1(\mathrm{mod}8)$ and lie outside the cases of Proposition prop:rat. The odd sizes not of the form $2^k+1$ and not congruent to $3$ or $5$ modulo $8$, and the even sizes $n\ge 6$, are not settled by the source or this module.

Two steps of that proof are recorded by the source as known. Section 2 of the source cites Kay for the fact that a symmetric tridiagonal Hamiltonian with perfect state transfer between its end vertices is persymmetric, and recalls that after a common shift the eigenvalues are then integers that alternate between even and odd. The module states the first for an arbitrary unitary commuting with the path Hamiltonian, and derives from the second the identity $P{\sum }_{i\in A}1/{\prod }_{j\ne i}(z_i-z_j)=1/2$ for the spectral class $A$ of a Hermitian matrix; the 2-adic valuation of that sum is the module’s own step.

The module D5/S3/Quantum/Dynamics/OddPathRationalWeightTransfer, with the spectral weights and integer moments at the middle vertex of D5/S3/Quantum/Dynamics/PathMiddleVertexMoments, proves the statement of the conjecture for every odd $n=2m+1\ge 3$, in both directions between the end vertices and with arbitrary real potentials: $n=3$ lies below the conjecture’s range, the sizes $n\equiv 3,5(\mathrm{mod}8)$ are the cases of Proposition prop:rat, and the sizes $n\equiv 1,7(\mathrm{mod}8)$ lie outside those cases. The even sizes $n\ge 6$ are not settled by the source or these modules. The source obtains its odd cases from the weight next to the middle vertex: with $S_1={\sum }_{r=1}^n(-1{)}^{r+n}{\alpha }_r$ and $S_2={\sum }_{r=1}^n(-1{)}^{r+n}{\alpha }_r^2$ for the ordered eigenvalues ${\alpha }_r$, its Corollary cor:middle gives, for $n$ odd, $r_{\frac{n-1}{2}}=\frac{\sqrt{S_2-S_1^2}}{2}$ and $q_{\frac{n+1}{2}}=S_1$; it also recalls, citing Cantoni and Butler, that the eigenvectors of a mirror-symmetric Hamiltonian are symmetric or antisymmetric. The modules use the diagonal entries of all the powers of the Hamiltonian at the middle vertex and the Hankel determinants of these moments of every size up to $m+1$.

Verified locator

• DOI: https://doi.org/10.1080/03081087.2018.1442810
• arXiv: https://arxiv.org/abs/1708.03283v2