Perturbative expansion of entanglement negativity

A common way to quantify entanglement in bipartite systems is through entanglement negativity. Because negativity is a non-analytic function of a density matrix, existing methods used in physics literature are insufficient to compute its derivatives. To this end we develop novel 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 new and 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.