Phase Transition in Iterated Quantum Protocols for Noisy Inputs

TL;DR

This paper investigates how iterated quantum protocols can exhibit phase transition-like behavior in their dynamical complexity, influenced by initial state purity, with implications for quantum information processing.

Contribution

It reveals that complex, fractal-like dynamics in quantum protocols undergo a phase transition driven by initial state purity, extending understanding of quantum chaos and nonlinear maps.

Findings

Sensitive, quasi-chaotic evolution occurs for mixed states.

A first order phase transition in dynamical behavior is identified.

Fractal structures in parameter space are linked to initial state purity.

Abstract

Quantum information processing exploits all the features quantum mechanics offers. Among them there is the possibility to induce nonlinear maps on a quantum system by involving two or more identical copies of the given system in the same state. Such maps play a central role in distillation protocols used for quantum key distribution. We determine that such protocols may exhibit sensitive, quasi-chaotic evolution not only for pure initial states but also for mixed states, i.e. the complex dynamical behavior is not destroyed by small initial uncertainty. We show that the appearance of sensitive, complex dynamics associated with a fractal structure in the parameter space of the system has the character of a first order phase transition. The purity of the initial state plays the role of the control parameter and the dimension of the fractal structure is independent of the purity value after passing the phase transition point. The critical purity coincides with the purity of a repelling fixed point of the dynamics and we show that all the pre-images of states from the close neighborhood of pure chaotic initial states have purity larger than this. Initial states from this set can be considered as quasi-chaotic.