  Recently, there has been substantial interest in studying the dynamics of
quantum theory beyond that of states, in particular, the dynamics of channels,
measurements, and higher-order transformations. Ref. [Phys. Rev. X 8(1), 011047
(2018)] pursues this using the process-matrix formalism, together with a
definition of the possible dynamics of such process matrices, and focusing
especially on the question of evolution of causal structures. One of its major
conclusions is a strong theorem saying that within the formalism, under
continuous and reversible transformations, the causal order between operations
must be preserved. Our result here challenges that of Ref. [Phys. Rev. X 8(1),
011047 (2018)]: if one is to take into account a full picture of the physical
evolution of operations within the standard quantum-mechanical formalism, then
the conclusion of Ref. [Phys. Rev. X 8(1), 011047 (2018)] does not hold. That
is, we show that under certain continuous and reversible dynamics, the causal
order between operations is not necessarily preserved. We moreover identify and
analyse the root of this apparent contradiction, specifically, that the
commonly accepted and widely applied framework of higher-order processes,
whilst mathematically sound, is not always appropriate for drawing conclusions
on physical dynamics. Finally, we show how to reconcile the elements of the
whole picture following the intuition based on entanglement processing by local
operations and classical communication.
