Instantaneous measurement can isolate the information

Consider a one-dimensional spin chain, from spin 1 to spin N, such that each spin interacts with its nearest neighbors. Performing a local operation (measurement) on spin N, we expect from the Lieb–Robinson velocity that, in general, the effect of this measurement achieves spin 1 after some while. But, in this paper, we show that if (a) the measurement on spin N is performed instantaneously and (b) the initial state of the spin chain is chosen appropriately, then the effect of the measurement on spin N never achieves spin 1. In other words, performing or not performing an instantaneous measurement on spin N at \(t=0\) does not alter the reduced dynamics of spin 1 for all the times \(t\ge 0\). We can interpret this as the following: The information of performing an instantaneous measurement on spin N is isolated such that it cannot achieve spin 1.