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Annalen der Physik

Witnessing Quantum Speedup through Mutual Information

Jing Wang, Yunan Wu, Song Shi-xue

Published Sep 26, 2019• 2 citations
Publisher

Abstract

Abstract Achieving dynamical speedup of evolution in an open quantum system plays a key role in many technological applications. However, how to detect quantum speedup is unclear. In this work, a method to witness quantum speedup through the measure of the mutual information is presented. It is shown that the speed of evolution of a quantum system, can be witnessed by calculating the mutual information variation, whose increase is a clear signature of dynamical speedup. The result is explained by considering the time evolution of two qubits under a one‐sided noisy channel, finding out that the mechanism for the quantum speedup is closely associated with the total exchange of information between the system and its environment, which can be expressed by the variation of mutual information. Quantitatively, the average speed of evolution is shown proportional to the average variation of the mutual information in an interval of time. The conclusion can not only explain why the quantum entanglement (or quantum coherence) and the classical correlation are neither necessary nor sufficient to speed up the quantum evolution, but also give a vital way of detecting quantum speedup in realistic environments.

DOI
10.1002/andp.201900225

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Abstract Achieving dynamical speedup of evolution in an open quantum system plays a key role in many technological applications. However, how to detect quantum speedup is unclear. In this work, a method to witness quantum speedup through the measure of the mutual information is presented. It is shown that the speed of evolution of a quantum system, can be witnessed by calculating the mutual information variation, whose increase is a clear signature of dynamical speedup. The result is explained by considering the time evolution of two qubits under a one‐sided noisy channel, finding out that the mechanism for the quantum speedup is closely associated with the total exchange of information between the system and its environment, which can be expressed by the variation of mutual information. Quantitatively, the average speed of evolution is shown proportional to the average variation of the mutual information in an interval of time. The conclusion can not only explain why the quantum entanglement (or quantum coherence) and the classical correlation are neither necessary nor sufficient to speed up the quantum evolution, but also give a vital way of detecting quantum speedup in realistic environments.
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.
We construct a general measure for detecting the quantum speedup in both closed and open systems. This 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 real dynamical speedup. To clarify the mechanisms of quantum speedup, we first introduce the concept of longitudinal and transverse types of speedup, and then apply the proposed measure to several typical closed and open quantum systems, illustrating that entanglement and the memory effect of the environment together can become resources for longitudinally or transversely accelerating dynamical evolution under certain conditions. Remarkably, a direct measurement of such speedup is feasible without the need for a tomographic reconstruction of the density matrix, which greatly enhances the feasibility of practical experimental tests.
We theoretically study the dynamical speedup of a quantum system in a nonequilibrium environment. Based on the trace distance, we derive the generalized Margolus-Levitin and Mandelstam-Tamm types of bounds on the quantum speed limit time of a quantum system evolving from an arbitrary initial state. We demonstrate that the mechanism for the speedup of dynamical evolution is closely associated with both the energy of the system and exchange of information between the system and its environment. It is shown that the nonequilibrium feature of the environment can speed up the quantum evolution in both Markovian and non-Markovian dynamics regions. We emphasize that the non-Markovian effect of the system dynamics is neither necessary nor sufficient to speed up the quantum evolution in a nonequilibrium environment.
Abstract The maximal evolution speed of any quantum system can be expressed by the quantum speed limit time. In this paper, we consider a model in which the system has a correlation with the environment. The influence of the initial correlation between the system and environment on the quantum speed limit is investigated. It is shown that the appearance of non-Markovianity effects causes the speedup of quantum evolution. Moreover, we demonstrate the dependence of quantum dynamical speedup on the quantum coherence of the correlated initial state.
The maximal evolution speed of a quantum system can be represented by quantum speed limit time (QSLT). We investigate the QSLT of a two-qubit system passing through a correlated channel (amplitude damping, phase damping, and depolarizing). By adjusting the correlation parameter of the channel and the initial entanglement, a method to accelerate the evolution speed of the system for some specific channels is proposed. It is shown that, in the amplitude damping channel and the depolarizing channel, QSLT may be shortened in some cases by increasing the correlation parameter of the channel and the initial entanglement, which are in sharp contrast to the phase damping channel. In particular, under depolarizing channels, the transition from no-speedup evolution to speedup evolution for the system can be realized by changing the correlation strength of the channel.
Quantum evolution can be accelerated in a non-Markovian environment. Previous results show that the formation of a system-environment bound state governs the quantum speedup. Although a stronger bound state in the system-environment spectrum may seem like it should cause greater speed of evolution, this seemingly intuitive thinking may not always be correct. We illustrate this by investigating a classical-driven qubit interacting with a photonic crystal waveguide in the presence of a mirror, resulting in non-Markovian dynamics for the system. Within the considered model, we show the influence of the mirror and the classical field on the evolution speed of the system. In particular, we find that the formation of a bound state is not the essential reason for the acceleration of evolution. The quantum speedup is attributed to the flow of information, regardless of the direction in which the information flows. Our conclusion can also be used in other non-Markovian environments.