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International Journal of Theoretical Physics

Quantum Speed Limit Under the Influence of Measurement-based Feedback Control

Lu Hou, Bin Shao, Chaoquan Wang

Published Feb 27, 2023• 4 citations
Publisher
DOI
10.1007/s10773-023-05318-8

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We investigate the quantum speed limit time (QSLT) of a two-level atom under quantum-jump-based feedback control or homodyne-based feedback control. Our results show that the two different feedback control schemes have different influences on the evolutionary speed. By adjusting the feedback parameters, the quantum-jump-based feedback control can induce speedup of the atomic evolution from an excited state, but the homodyne-based feedback control cannot change the evolutionary speed. Additionally, the QSLT for the whole dynamical process is explored. Under the quantum-jump-based feedback control, the QSLT displays oscillatory behaviors, which implies multiple speed-up and speed-down processes during the evolution. While, the homodyne-based feedback control can accelerate the speed-up process and improve the uniform speed in the uniform evolution process.
We investigate the probability that a conditioned quantum trajectory exceeds an ensemble-derived quantum speed limit (QSL) velocity under continuous measurement-based feedback. Using the finite-time average fidelity velocity, we derive an upper bound on the exceedance probability in terms of its mean and variance. We further obtain state-independent bounds from uniform constraints on the fidelity drift and diffusion, including an exponential concentration bound. For a feedback-controlled qubit, we derive explicit model-specific bounds and show that they are nontrivial in a short-time regime. We also identify a finite-time regime in which the exceedance probability vanishes exactly due to the restricted support of the conditioned dynamics. Monte Carlo simulations confirm the analytical bounds and illustrate their tightness.
Fundamental trade-off relations, such as quantum speed limit and quantum thermodynamic uncertainty relation, describe the performance limits of quantum systems by imposing that improvements in speed or precision necessitate a substantial thermodynamic cost. Quantum feedback control, which is a pivotal technique for manipulating quantum dynamics based on measurement outcomes, is widely employed to enhance system performance. Nevertheless, its impact on these fundamental bounds remains an open question. This work elucidates this influence by establishing a theoretical framework for quantum speed limit and quantum thermodynamic uncertainty relation under a paradigmatic Markovian feedback protocol. We derive general inequalities incorporating the effects of feedback control on speed and precision. Through numerical simulations on a simple two-level system and quantum error correction, a key application of quantum feedback control, we validate our derived bounds and demonstrate that feedback control can indeed improve both speed and precision beyond those achievable limits in uncontrolled systems. Next, to elucidate the mechanism behind these improvements and the qualitative difference from uncontrolled dynamics, we analyze the governing thermodynamic costs, which are the fundamental quantities that constrain speed and precision, within a simple model. Our analysis reveals that feedback can improve the time scaling order of these costs. This modification of the dynamical scaling is the origin of the qualitative performance gain, signifying that the feedback-induced improvements of performance are not merely quantitative but represent a fundamental shift. Consequently, our work offers a comprehensive understanding of how feedback control impacts the fundamental limits on the speed and precision of quantum systems, providing crucial insights for designing high-performance quantum technologies.