Abstract
Although a complete picture of the full evolution of complex quantum systems would certainly be the most desirable goal, for particular Quantum Information Processing schemes such an analysis is not necessary. When quantum correlations between only specific elements of a many-body system are required for the performance of a protocol, a more distinguished and specialised investigation is helpful. Here, we provide a striking example with the achievement of perfect state transfer in a spin chain without state initialisation, whose realisation has been shown to be possible in virtue of the correlations set between the first and last spin of the transmission-chain.
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The final state of spin N can be easily obtained from Eqs. (3)-(5). For instance, if N is even, \(\langle{\hat{Z}_N(t^*)}\rangle=\langle{\hat{Z}_1(0)}\rangle\), \(\langle{\hat{X}_1\hat{X}_N(t^*)}\rangle=\langle{\hat{X}_1\hat{X}_N(0)}\rangle\) and \(\langle{\hat{X}_1\hat{Y}_N(t^*)}\rangle=\langle{\hat{Y}_1\hat{X}_N(0)}\rangle\). If qubit N has been projected onto ∣ ± N 〉 [for which \(\langle{\hat{X}_N(0)}\rangle=\pm(-1)^{\frac{N}{2}}\)], we have \(\langle{\hat{Z}_N(t^*)}\rangle=\langle{\hat{Z}_1(0)}\rangle\), \(\langle{\hat{X}_1\hat{X}_N(t^*)}\rangle=\pm(-1)^{\frac{N}{2}}\langle{\hat{X}_1(0)}\rangle\) and \(\langle{\hat{X}_1\hat{Y}_N(t^*)}\rangle=\pm(-1)^{\frac{N}{2}}\langle{\hat{Y}_1(0)}\rangle\). The state of spin N, after the measurement performed on spin 1, will satisfy \(\langle{\hat{Z}_N(t^*)}\rangle=\langle{\hat{Z}_1(0)}\rangle\), \(\langle{\hat{X}_N(t^*)}\rangle=(-1)^{\frac{N}{2}}c\,\langle{\hat{X}_1(0)}\rangle\) and \(\langle{\hat{Y}_N(t^*)}\rangle=(-1)^{\frac{N}{2}}c\,\langle{\hat{Y}_1(0)}\rangle\), where c is the product of the measurement outcomes at 1 (after the evolution) and N (before the evolution). The state \((\hat{Z}^{\frac{N}{2}})\rho^{in}(\hat{Z}^{\frac{N}{2}})\) [\((\hat{Z}^{\frac{N}{2}+1})\rho^{in}(\hat{Z}^{\frac{N}{2}+1})\)] satisfies these conditions for c = 1 (c = − 1)
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Di Franco, C., Paternostro, M., Kim, M.S. (2011). Bypassing State Initialisation in Perfect State Transfer Protocols on Spin-Chains. In: van Dam, W., Kendon, V.M., Severini, S. (eds) Theory of Quantum Computation, Communication, and Cryptography. TQC 2010. Lecture Notes in Computer Science, vol 6519. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-18073-6_14
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