  One of the basic lessons of quantum theory is that one cannot obtain
information on an unknown quantum state without disturbing it. Hence, by
performing a certain measurement, we limit the other possible measurements that
can be effectively implemented on the original input state. It has been
recently shown that one can implement sequentially any device, either channel
or observable, which is compatible with the first measurement [T. Heinosaari
and T. Miyadera: Phys. Rev. A Vol 91 (2015), 022110]. In this work we prove
that this can be done, apart from some special cases, only when the succeeding
device is implemented on a larger system than just the input system. This means
that some part of the still available quantum information has been flown to the
environment and cannot be gathered by accessing the input system only. We
characterize the size of the post-measurement system by determining the class
of measurements for the observable in question that allow the subsequent
realization of any measurement process compatible with the said observable. We
also study the class of measurements that allow the subsequent realization of
any observable jointly measurable with the first one and show that these two
classes coincide when the first observable is extreme.
