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arXiv:2408.17442

Analysis on the von Neumann entropy under the measurement-based feedback control

Kohei Kobayashi

Published Aug 30, 2024• 0 citations
arXivHTMLPDF

Abstract

The measurement-based feedback (MBF) control offers several powerful means for preparing the desired target quantum state. Therefore, it is important to investigate fundamental properties of MBF. In particular, how the entropy of the controlled system under the MBF behaves is of great interest. In this study, we examine this problem by deriving a sufficient condition that the time derivative of the von Neumann entropy is nonnegative under the MBF control. This result is rigorously characterized by the variance of the system observable and the quantumness of a given decoherence. We show the validity of the result and physical interpretation in the example of qubit stabilizing.

DOI
10.48550/arxiv.2408.17442

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Control of open quantum dynamics is of great interest for realizing quantum technologies. Therefore, it is an important task to quantify and characterize the entropy for open quantum systems under decoherence. In this paper, we study the time evolution of the von Neumann entropy for open quantum systems described by the Lindblad master equation. Note that, in particular, when the decoherence corresponds to the measurement for the observable in the system, the von Neumann entropy tends to monotonically increases as the variance becomes larger. Furthermore, we present a lower bound of the von Neumann entropy in the long-time limit. This lower bound has advantages of being straightforwardly calculated and applicable to a general Markovian open quantum system.
By introducing the von Neumann entropy as a measure of the extent of noise, this paper discusses the entropy evolution in a two-level quantum feedback controlled system. The results show that the feedback control can induce the reduction of the degree of noise, and different control schemes exhibit different noise controlling ability, the extent of the reduction also related with the position of the target state on the Bloch sphere. It is shown that the evolution of entropy can provide a real time noise observation and a systematic guideline to make reasonable choice of control strategy.
In this paper we investigate some general properties of equilibrium points of finite quantum dynamics which provide fundamental results for designing a controller and a measurement apparatus. Firstly, the averaged quanturn dynamics is shown to be stable in the Lyapunov sense. This implies that the average behavior of the quantum dynamics is characterized by the set of the equilibrium points. Secondly, an explicit form of a unique equilibrium point associated with the controller and the measurement apparatus, which provides a practical procedure to attain a desired equilibrium point, is shown. A quantum state can be utilized as an information source quantified by “von Neumann entropy”. We derive a necessary and sufficient condition for the entropy of the equilibrium points to be maximum (or minimum).
We derive a universal inequality that provides a lower bound on the ensemble-averaged von Neumann entropy change in a quantum system subject to continuous measurement and dissipation. Our result clarifies how entropy production is fundamentally constrained by three distinct contributions: (i) the non-Hermitian structure of the dissipation operator, (ii) the standard variance associated with measurement-induced fluctuations, and (iii) a generalized quantum variance reflecting the noncommutativity between the measurement observable and the quantum state. This third term vanishes when the state and observable commute, and thus represents a purely quantum contribution arising from coherence disturbance and measurement backaction. The derived inequality generalizes classical information-thermodynamic relations, such as the Sagawa--Ueda inequality, to the quantum regime, providing a new perspective on the trade-offs between information acquisition, control, and entropy production in continuously monitored open quantum systems.
In a quantum feedback control loop, it is often confronted with time delays mainly originated from the transition delay of signals, which may cause quantum feedback control systems unstable. The purpose of this paper is to deal with a problem of quantum measurement-based feedback control systems subject to feedback-loop time delays. A physical model for a quantum measurement-based feedback control system with time delay is presented for the H∞control. Fundamental properties such as dissipation, gain and stability for this model are presented and characterized using linear matrix inequalities (LMIs). A numerical procedure is proposed for H∞controller designs for our model. An example is given to illustrate our numerical procedure.
It is notoriously known that in contrast to the measure of classical macro systems, the quantum measurement will in general cause quantum state collapse. In this note, we will present some further results on stabilizing control of quantum systems with measurement being involved. To be precise, we will study the effects of the open-loop control and the feedback control in preparing an arbitrarily desired eigenstate of an effective Hamiltonian, respectively. For the open-loop control case, it is shown that no matter how to select the measurement channel, control channel and the control law, one cannot prepare all the eigenstates of the effective Hamiltonian with arbitrary high fidelity for all sufficient large time. While for the feedback case, we not only demonstrate how to select the measurement channel properly to achieve the control target, but also show if not, what cannot be done by the measurement-based feedback control.