Pretty Good State Transfer in Qubit Chains - The Heisenberg Hamiltonian
Abstract
Pretty good state transfer in networks of qubits occurs when a continuous-time quantum walk allows the transmission of a qubit state from one node of the network to another, with fidelity arbitrarily close to 1. We prove that in a Heisenberg chain with n qubits there is pretty good state transfer between the nodes at the j-th and (n-j+1)-th position if n is a power of 2. Moreover, this condition is also necessary for j=1. We obtain this result by applying a theorem due to Kronecker about Diophantine approximations, together with techniques from algebraic graph theory.
Keywords
Cite
@article{arxiv.1608.04722,
title = {Pretty Good State Transfer in Qubit Chains - The Heisenberg Hamiltonian},
author = {Leonardo Banchi and Gabriel Coutinho and Chris Godsil and Simone Severini},
journal= {arXiv preprint arXiv:1608.04722},
year = {2018}
}
Comments
The first version of this paper made available on Arxiv contained the false claim that pretty good state transfer occurs between end vertices in a path of length $n$ where $n$ is a prime congruent to $1$ modulo $4$. This claim has been corrected in this version