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Pretty good state transfer in qubit chains – the Heisenberg Hamiltonian. (English) Zbl 1359.81047
Summary: 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 positions 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.{