  At the onset of an interaction between two initially independent systems,
each system tends to experience an increase in its n-Renyi entropies, such as
its von Neumann entropy (n = 1) and its mixedness (n = 2). We here ask which
properties of a system determine how quickly its Renyi entropies increase and,
therefore, how sensitive the system is to becoming entangled. We find that the
rate at which the n-Renyi entropy increases in an interaction is determined by
a quantity which we term the n-fragility of the system. The 2-fragility is
closely related to the notion of 2-norm coherence, in that it too quantifies
the extent to which a density matrix is off-diagonal with respect to the
eigenbasis of a reference operator. Nevertheless, the 2-fragility is not a
coherence monotone in the resource theoretic sense since it depends also on the
eigenvalues of the reference operator. It is this additional sensitivity to the
eigenvalues of the reference operator, here the interaction Hamiltonian, which
enables the 2-fragility to quantify the rate of entropy production in
interactions. We give an example using the light-matter interaction and we
anticipate applications to the study of the rates at which two systems exchange
classical and quantum information when starting to interact.
