Detecting quantum speedup in closed and open systems

Zhen-Yu Xu

doi:10.1088/1367-2630/18/7/073005

Abstract

We construct a general measure for detecting the quantum speedup in both closed and open systems. The speed measure is based on the changing rate of the position of quantum states on a manifold with appropriate monotone Riemannian metrics. Any increase in speed is a clear signature of dynamical speedup. To clarify the mechanisms for quantum speedup, we first introduce the concept of longitudinal and transverse types of speedup: the former stems from the time evolution process itself with fixed initial conditions, while the latter is a result of adjusting initial conditions. We then apply the proposed measure to several typical closed and open quantum systems, illustrating that quantum coherence (or entanglement) and the memory effect of the environment together can become resources for longitudinally or transversely accelerating dynamical evolution under specific conditions and assumptions.

Highlights

The speed of quantum evolution determines how long it will take for a quantum state to evolve to a target state in a given process
Further studies have shown that the memory effect of the environment can induce dynamical acceleration in open quantum systems [31, 36]; this phenomenon has been experimentally observed in recent studies with a cavity field as the open system and the number of atoms as the controllable environment [32]
The above discoveries are of considerable interest and importance because they reveal that both entanglement and the memory effect may be beneficial for controlling dynamical processes

Summary

INTRODUCTION

The speed of quantum evolution determines how long it will take for a quantum state to evolve to a target state in a given process. From another point of view, we may focus on the QSL time itself, with the actual driving time τ held fixed In this case, τ /τQSL = 1 implies that the evolution has already been proceeding. To further clarify the mechanisms for the quantum speedup, the “speedup” is divided into two types, i.e., longitudinal speedup and transverse speedup The former arises during the evolution process itself with fixed initial conditions, while the latter is due to a change in the initial conditions. The memory effect in the given examples is confirmed to be a subtle but important factor in determining the longitudinal or transverse acceleration of the evolution process. The mechanism for longitudinal and transverse quantum speedup in closed and open systems is analyzed in Sec. III with two typical and pedagogical examples.

DETECTION OF THE SPEEDUP OF DYNAMICAL EVOLUTION

Two-qubit case

Single-qubit case

Open systems: a physical model of two-level atoms coupled to leaky cavities

CONCLUSION