  Negativity is an entanglement monotone frequently used to quantify
entanglement in bipartite states. Because negativity is a non-analytic function
of a density matrix, existing methods used in the physics literature are
insufficient to compute its derivatives. To this end we develop techniques in
the calculus of complex, patterned matrices and use them to conduct a
perturbative analysis of negativity in terms of arbitrary variations of the
density operator. The result is an easy-to-implement expansion that can be
carried out to all orders. On the way we provide convenient representations of
the partial transposition map appearing in the definition of negativity. Our
methods are well-suited to study the growth and decay of entanglement in a wide
range of physical systems, including the generic linear growth of entanglement
in many-body systems, and have broad relevance to many functions of quantum
states and observables.
