reaches the receiver site and fidelity peaks for the first time. Though the later peaks might be larger it is physically unwise to wait for such long times as in practical cases the interaction with environment and its induced decoherence deteriorates the quality of transmission. So that we focus on the first peak at which the fidelity takes its maximal value, i.e. $F_{max}^{\pm}=F_u^{\pm}(t_{opt})$.

In Figs. \[fig2\](b) and (c) the maximal fidelities $F_{max}^{+}$ and $F_{max}^{-}$ are plotted versus length $N$. As it is clear from these figures the fidelities are both high and go down almost linearly with very small slopes. A linear fit to data shows that $F_{max}^{+}=-0.007N+1.024$ and $F_{max}^{-}=-0.005N+1.016$. One can use these linear fits to extrapolate the fidelities in longer chains which shows that for chains up to $N \sim 50$ the fidelities are still above the classical threshold 2/3. This indeed shows the very high potential of this strategy for quantum state transfer across a many-body system. In Fig. \[fig2\](d) the optimal time $t_{opt}$ is plotted versus $N$ which also shows a linear dependence on $N$.

Quantum State Transfer: Restricted Basis {#sec4}
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Very often due to practical issues it is not possible to accomplish quantum measurement 