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K. Wootters, Purification of noisy entanglement and faithful teleportation via noisy channels, Phys. Rev. Lett. 78(10), 2031 (1997) 20. H. P. A. K. Wootters, Mixed-state entanglement and quantum error correction. Phys. Rev. A 54(5), 3824–3851 (1996) 30 2 Exploiting Quench Dynamics in Spin Chains 21. J. A. Nielsen, Entanglement in a simple quantum phase transition. Phys. Rev. A 66(3), 032110 (2002) 22. -P. Breuer, F. Petruccione, The theory of open quantum systems. (Clarendon Press, Oxford, UK, 2002) 23.

In general, convenient to express q ^ on the state space of the chain composed of N spin-1=2 particles any operator A may be decomposed into ^¼ A 22N X h i ^k ^ ky X ^X Tr q ð2:10Þ k¼1 h i À Á ^k ; X ^ l Tr X ^ l ¼ dk;l ^ yX X k ð2:11Þ 20 2 Exploiting Quench Dynamics in Spin Chains X ^k ¼ 1 X ð2:12Þ k which is sometimes referred to as Hilbert–Schmidt operator decomposition [22]. For an operator A defined on the Hilbert space spanned by the two spin-12 degrees of freedom associated to site l of a spin chain configuration it is convenient to choose the following set of orthogonal operators: n o ^ ¼ r ^" ^# ^þ ^À l ¼ 1; 2; .

However, our data for Tmax in a quench N suggesting that the process of D : 1 ! 0 are seen to obey the scaling Tmax $ pJ establishing entanglement is slightly slower than this estimate. Yet, the linear scaling with N is otherwise consistent with the causal cone picture. This delay hints towards a collective effect, leading to an optimal entanglement as a result of many such local events. The excitation that is emitted from the centre of the chain (meaningful in the case of odd N) will, at the optimal time Tmax , have encountered the terminal sites and will have been reflected from the boundaries.