Figure 4

The amplification efficiency quantified by the quantum Chernoff information.

Here, the amplification is taken relative to a reference case, . The QCB predicts that the relative efficiency is with ξ_QCB from Eq. (19) (black, dotted lines). The quadratic growth of the redundancy, given by Eq. (29), is shown as black, solid lines. For all data, the QCB prediction and quadratic growth match well with the numerically computed results. For very small relative efficiencies there is some deviation, which is due to the finite obtainable numerically. The full QCB result deviates from quadratic behavior for very long times (i.e., on the order of the recurrence time of an individual spin interaction, which goes as ). The data in the figure is as follows: The five lines are for varying haziness (the initial entropy h = H[(1 + a)/2] of a single environment spin^32,33) h = 0, 1/5, 2/5, 3/5 and 4/5 from top to bottom. Each set of symbols shows the numerical result for for three initially pure states of the system ( = 1/2, 1/8 and 1/32) relative to the QCB reference case with with h = 0 and t = π/8. Further details can be found in the Methods.