Pretty Good State Transfer in Qubit Chains - The Heisenberg Hamiltonian

TL;DR

This paper proves that in a Heisenberg qubit chain with a power-of-two number of qubits, pretty good state transfer occurs between symmetric nodes, using algebraic and number theory techniques.

Contribution

It establishes necessary and sufficient conditions for pretty good state transfer in Heisenberg chains with power-of-two qubits, linking quantum information transfer to Diophantine approximation.

Findings

Pretty good state transfer occurs when n is a power of 2.

Necessary condition for j=1 is n being a power of 2.

Uses algebraic graph theory and Diophantine approximation techniques.

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.