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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.{
©2017 American Institute of Physics}

MSC:
81P45 Quantum information, communication, networks (quantum-theoretic aspects)
68M10 Network design and communication in computer systems
60G50 Sums of independent random variables; random walks
82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
11K60 Diophantine approximation in probabilistic number theory
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