Competition between decoherence and purification: Quaternionic representation and quaternionic fractals

Abstract

We consider the competition between decoherence processes and an iterated quantum purification protocol. We show that this competition can be modelled by a nonlinear map onto the quaternion space. This nonlinear map has complicated behaviours, inducing a fractal border between the area of the quantum states dominated by the effects of the purification and the area of the quantum states dominated by the effects of the decoherence. The states on the border are unstable. The embedding in a 3D space of this border is like a quaternionic Julia set or a Mandelbulb with a fractal inner structure.Qubits are the resource of the quantum information as bits for the classical information, and are the main subject for future technologies as quantum computers. In contrast with bits, qubits exhibit states which are impossible at a classical level as Schrödinger cat states (the qubit is in a superposition of 0 and 1). These purely quantum properties are the resource to drastically increase the performance of the computing. But the noises of the environment generate a physical process called decoherence which suppresses the purely quantum properties. There is a protocol, called purification, which permits to restore the quantum behaviour. The result of the competition between a permanent decoherence process and a repeated purification protocol is not simple because this generates a chaotic process. We show that this one is a generalisation of the famous Julia map (which generates the famous fractals known as the Julia and the Mandelbrot sets). More precisely, in place of a map of the complex plane, the decoherence-purification competition map is a map of the quaternionic space (so-called Hamilton's number set, which are numbers which do not commute, i.e. zw=wz with z and w two quaternions). The decoherence-purification map generates 3D fractal sets similar to a Mandelbulb (3D generalisation of a Mandelbrot set) with a fractal inner structure.