Quantum / CONCEPT

Early state exclusion in seven-site transfer chains with symmetric spectrum

Escobar, Garcia and Minenkova (arXiv:2507.18767, Section 4) conjecture 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, and they prove two one-parameter families of spectra. The conjecture is true: for every chain of seven sites with positive couplings, real potentials and a spectrum closed under negation, and every time T > 0 at which the chain has perfect state transfer between its end sites and before which it has none, the first-site amplitude has no zero in the open interval (0, T) if and only if every positive eigenvalue is an integer multiple of the smallest positive eigenvalue.