Abstract
We review the dual-rail encoding (Burgarth and Bose, Phys Rev A 71:052315, 2005) which demonstrates how the problem of dispersion in quantum state transfer in spin chain communication can be attacked and overcome through performing measurements at the receiver side. We discuss the performance of the dual-rail technique in detail with respect to noise, disorder in the chain couplings (Burgarth and Bose, New J Phys 7:135, 2005) and deviations from a strict one-dimensionality. We then show how the dual-rail method can be made more efficient by using multiple channels (Burgarth et al., Int J Quant Inf 4:405, 2006; J Phys A Math Gen 38:6793, 2005). We provide a convergence theorem which shows that any nearest-neighbor excitation preserving chain is capable of efficient and perfect state transfer using a multi-rail encoding.
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Notes
- 1.
Specifically, we identify the vector \(\left \vert \boldsymbol{0}\right \rangle _{i}\) with the factorized state where all the qubits of the chain are initialized in |0⟩, while \(\left \vert \boldsymbol{n}\right \rangle _{i}\) with the factorized state where all the qubits of the chain are in zero apart from the n-th one which is in \(\vert 1\rangle\).
- 2.
This is not a strong assumption. If the excitation was fully randomly distributed, the probability would scale as N −1. By searching for good arrival times, this can be slightly increased to N −2∕3. 
- 3.
Notice that strictly speaking the eigenvectors of the Hamiltonian are not the same as those of the time evolution operators. The latter still can have evolution times at which additional degeneracy can increase the set of eigenstates. A trivial example is given for t = 0 where all states become eigenstates. But it is always possible to find times t at which the eigenstates of U(t) coincide with those of H.
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Burgarth, D.K., Giovannetti, V. (2014). Dual- and Multi-rail Encoding. In: Nikolopoulos, G., Jex, I. (eds) Quantum State Transfer and Network Engineering. Quantum Science and Technology. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-39937-4_3
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