A short tutorial on Wirtinger Calculus with applications in quantum information

TL;DR

This tutorial introduces Wirtinger Calculus as an effective tool for optimizing complex matrix parameters, with applications demonstrated in quantum information theory.

Contribution

It provides a clear pedagogical introduction to Wirtinger Calculus and illustrates its usefulness in quantum information applications.

Findings

Simplifies optimization of complex matrix parameters

Demonstrates applications in quantum information theory

Provides a pedagogical framework for Wirtinger Calculus

Abstract

The optimization of system parameters is a ubiquitous problem in science and engineering. The traditional approach involves setting to zero the partial derivatives of the objective function with respect to each parameter, in order to extract the optimal solution. However, the system parameters often take the form of complex matrices. In such situations, conventional methods become unwieldy. The `Wirtinger Calculus' provides a relatively simple methodology for such optimization problems. In this tutorial, we provide a pedagogical introduction to Wirtinger Calculus. To illustrate the utility of this framework in quantum information theory, we also discuss a few example applications.