Perturbative expansion of entanglement negativity using patterned matrix calculus

TL;DR

This paper introduces a novel perturbative expansion technique for negativity, an entanglement measure, using patterned matrix calculus, enabling detailed analysis of entanglement variations in quantum systems.

Contribution

It develops a new mathematical framework for computing derivatives of negativity, facilitating perturbative analysis to all orders, which was previously challenging due to its non-analytic nature.

Findings

Provides a practical expansion method for negativity derivatives

Offers new representations of the partial transposition map

Applicable to study entanglement dynamics in many-body systems

Abstract

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.