Exponential Sensitivity and its Cost in Quantum Physics

TL;DR

This paper demonstrates that state selective quantum protocols exhibit exponential sensitivity to initial states, leading to chaotic behavior and a quantum Schrödinger microscope, with fundamental limits on the number of copies needed.

Contribution

It shows that exponential sensitivity is generic in state selective protocols and establishes a bound on the ensemble size reduction in quantum chaos.

Findings

Chaotic behavior is common in state selective protocols.

Any complex rational polynomial map can be realized in quantum systems.

Exponential sensitivity requires exponential reduction in initial ensemble size.

Abstract

State selective protocols, like entanglement purification, lead to an essentially non-linear quantum evolution, unusual in naturally occurring quantum processes. Sensitivity to initial states in quantum systems, stemming from such non-linear dynamics, is a promising perspective for applications. Here we demonstrate that chaotic behaviour is a rather generic feature in state selective protocols: exponential sensitivity can exist for all initial states in an experimentally realisable optical scheme. Moreover, any complex rational polynomial map, including the example of the Mandelbrot set, can be directly realised. In state selective protocols, one needs an ensemble of initial states, the size of which decreases with each iteration. We prove that exponential sensitivity to initial states in any quantum system have to be related to downsizing the initial ensemble also exponentially. Our results show that magnifying initial differences of quantum states (a Schr\"odinger microscope) is possible, however, there is a strict bound on the number of copies needed.