Quantum / CONCEPT

No perfect state transfer at time pi on rational paths with 2m vertices, m = 2 mod 4 or m = 3, 5 mod 8

Kirkland, McLaren, Pereira, Plosker and Zhang (arXiv:1708.03283) conjecture that a weighted path on at least four vertices, with or without potentials, whose edge weights are all rational has no perfect state transfer between its end vertices at readout time pi, and prove it for n = 4 and for n congruent to 3 or 5 modulo 8. The statement holds for every even number of vertices n = 2m with m congruent to 2 modulo 4 or to 3 or 5 modulo 8, that is for n congruent to 4 modulo 8 or to 6 or 10 modulo 16, with arbitrary real potentials and in both directions. The conjecture is not settled for n congruent to 0 modulo 8 or to 2 or 14 modulo 16.