Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Transactions of the Society of Instrument and Control Engineers

Analysis of Equilibrium Points of Quantum Controlled Dynamics

Naoki Yamamoto, Koji Tsumura, Shinji Hara

Published Jan 1, 2004• 0 citations
Publisher

Abstract

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).

DOI
10.9746/sicetr1965.40.693

Related papers

Analysis of Lyapunov Method for Control of Quantum States

Xiaoting Wang, S. G. Schirmer

The natural trajectory tracking problem is studied for generic quantum states represented by density operators. A control design based on the Hilbert-Schmidt distance as a Lyapunov function is considered. The control dynamics is redefined on an extended space where the LaSalle invariance principle can be correctly applied even for non-stationary target states. LaSalle's invariance principle is used to derive a general characterization of the invariant set, which is shown to always contain the critical points of the Lyapunov function. Critical point analysis of the latter is used to show that, for generic states, it is a Morse function with n! isolated critical points, including one global minimum, one global maximum and n!-2 saddles. It is also shown, however, that the actual dynamics of the system is not a gradient flow, and therefore a full eigenvalue analysis of the linearized dynamics about the critical points of the dynamical system is necessary to ascertain stability of the critical points. This analysis shows that a generic target state is locally asymptotically stable if the linearized system is controllable and the invariant set is regular, and in fact convergence to the target state (trajectory) in this case is almost global in that the stable manifolds of all other critical points form a subset of measure zero of the state space. On the other hand, if either of these sufficient conditions is not satisfied, the target state ceases to be asymptotically stable, a center manifold emerges around the target state, and the control design ceases to be effective.

arXiv:0801.0702•IEEE Transactions on Automatic Control•Feb 17, 2010•152 citations

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

Kohei Kobayashi

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.

arXiv:2408.17442•Aug 30, 2024•0 citations

Dynamics of quantum entropy maximization for finite-level systems

Anton A. Selivanov

The problem of constructing models for the statistical dynamics of finite-level quantum mechanical systems is considered. The maximum entropy principle formulated by E.T. Jaynes in 1957 and asserting that the entropy of any physical system increases until it attains its maximum value under constraints imposed by other physical laws is applied. In accordance with this principle, the von Neumann entropy is taken for the objective function; a dynamical equation describing the evolution of the density operator in finite-level systems is derived by using the speed gradient principle. In this case, physical constraints are the mass conservation law and the energy conservation law. The stability of the equilibrium points of the system thus obtained is investigated. By using LaSalle’s theorem, it is shown that the density function tends to a Gibbs distribution, under which the entropy attains its maximum. The method is exemplified by analyzing a finite system of identical particles distributed between cells. Results of numerical simulation are presented.

Vestnik St Petersburg University Mathematics•Dec 1, 2011•2 citations

THEORY OF THE CONTROL OF OBSERVABLE QUANTUM SYSTEMS

V. P. Belavkin

Abstract. The fundamental mathematical definitions of the controlled feedback Markov dynamics of quantum-mechanical systems are introduced with regard to the dynamical reduction and filtering of quantum states in the course of quantum measurement in either discrete or continuous real time. The concept of sufficient coordinates for the description of a posteriori quantum states in a given class is introduced and it is proved that they form a classical Markov process with values in either state operators or state vector space. The general problem of optimal control of a quantum-mechanical system is discussed and the corresponding Bellman equation in the space of sufficient coordinates is derived. The results are illustrated in the example of control of the semigroup dynamics of a quantum system that is instantaneously observed at discrete times and evolves between measurement times according to the Schrődinger equation. 1.

•Jan 1, 2005•56 citations

Localizing quantum information

Fahimeh Bayeh, Andre Kornell

The idea that information can be localized is pervasive in physics. In thermodynamics, entropy flows from one reservoir to another, and in quantum gravity, the entropy content of some spatial regions is bounded. In quantum information theory, information is transmitted from one place to another, and entanglement can provide an advantage when there are constraints on such transmission. Shannon entropy quantifies localizable information in the sense that marginalization defines an outer measure on the set of parts of a classical multipartite system. In contrast, von Neumann entropy does not quantify localizable information in this sense. However, a variant of von Neumann entropy, which originates in noncommutative geometry, does. We prove that this adjusted von Neumann entropy is the minimum quantum entropy that quantifies localizable information. We also prove that this is the unique quantum entropy that characterizes Bell states in terms of redundant information, directly generalizing the classical case. We work with finite-dimensional $C^*$-algebras throughout, modeling finite quantum systems that may have superselection sectors.

quant-pharXiv:2609.01966•Sep 3, 2026•0 citations

The characteristics of limit points of any trajectory of mixed-state quantum systems in the Lyapunov-based control design

Sen Kuang, Shuang Cong

For mixed-state quantum systems without degenerate transition, this paper chooses the expectation value of a mechanical quantity as a Lyapunov function, designs the control laws that guarantee the monotonic decreasing of the Lyapunov function, and mainly studies the characteristics of limit points of any system trajectory under the action of control fields. Research results show that for a fully connected system, the limit point of any trajectory is one of its equilibrium states while for an only connected system, the limit points of any trajectory is not isolated. In the latter case, this paper also gives the strict expression of the limit points via the Bloch vector framework of density operators.

Chinese Control Conference (CCC)•Jul 29, 2010•0 citations
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).
The natural trajectory tracking problem is studied for generic quantum states represented by density operators. A control design based on the Hilbert-Schmidt distance as a Lyapunov function is considered. The control dynamics is redefined on an extended space where the LaSalle invariance principle can be correctly applied even for non-stationary target states. LaSalle's invariance principle is used to derive a general characterization of the invariant set, which is shown to always contain the critical points of the Lyapunov function. Critical point analysis of the latter is used to show that, for generic states, it is a Morse function with n! isolated critical points, including one global minimum, one global maximum and n!-2 saddles. It is also shown, however, that the actual dynamics of the system is not a gradient flow, and therefore a full eigenvalue analysis of the linearized dynamics about the critical points of the dynamical system is necessary to ascertain stability of the critical points. This analysis shows that a generic target state is locally asymptotically stable if the linearized system is controllable and the invariant set is regular, and in fact convergence to the target state (trajectory) in this case is almost global in that the stable manifolds of all other critical points form a subset of measure zero of the state space. On the other hand, if either of these sufficient conditions is not satisfied, the target state ceases to be asymptotically stable, a center manifold emerges around the target state, and the control design ceases to be effective.
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.
The problem of constructing models for the statistical dynamics of finite-level quantum mechanical systems is considered. The maximum entropy principle formulated by E.T. Jaynes in 1957 and asserting that the entropy of any physical system increases until it attains its maximum value under constraints imposed by other physical laws is applied. In accordance with this principle, the von Neumann entropy is taken for the objective function; a dynamical equation describing the evolution of the density operator in finite-level systems is derived by using the speed gradient principle. In this case, physical constraints are the mass conservation law and the energy conservation law. The stability of the equilibrium points of the system thus obtained is investigated. By using LaSalle’s theorem, it is shown that the density function tends to a Gibbs distribution, under which the entropy attains its maximum. The method is exemplified by analyzing a finite system of identical particles distributed between cells. Results of numerical simulation are presented.
Abstract. The fundamental mathematical definitions of the controlled feedback Markov dynamics of quantum-mechanical systems are introduced with regard to the dynamical reduction and filtering of quantum states in the course of quantum measurement in either discrete or continuous real time. The concept of sufficient coordinates for the description of a posteriori quantum states in a given class is introduced and it is proved that they form a classical Markov process with values in either state operators or state vector space. The general problem of optimal control of a quantum-mechanical system is discussed and the corresponding Bellman equation in the space of sufficient coordinates is derived. The results are illustrated in the example of control of the semigroup dynamics of a quantum system that is instantaneously observed at discrete times and evolves between measurement times according to the Schrődinger equation. 1.
The idea that information can be localized is pervasive in physics. In thermodynamics, entropy flows from one reservoir to another, and in quantum gravity, the entropy content of some spatial regions is bounded. In quantum information theory, information is transmitted from one place to another, and entanglement can provide an advantage when there are constraints on such transmission. Shannon entropy quantifies localizable information in the sense that marginalization defines an outer measure on the set of parts of a classical multipartite system. In contrast, von Neumann entropy does not quantify localizable information in this sense. However, a variant of von Neumann entropy, which originates in noncommutative geometry, does. We prove that this adjusted von Neumann entropy is the minimum quantum entropy that quantifies localizable information. We also prove that this is the unique quantum entropy that characterizes Bell states in terms of redundant information, directly generalizing the classical case. We work with finite-dimensional C∗C^*C∗-algebras throughout, modeling finite quantum systems that may have superselection sectors.
For mixed-state quantum systems without degenerate transition, this paper chooses the expectation value of a mechanical quantity as a Lyapunov function, designs the control laws that guarantee the monotonic decreasing of the Lyapunov function, and mainly studies the characteristics of limit points of any system trajectory under the action of control fields. Research results show that for a fully connected system, the limit point of any trajectory is one of its equilibrium states while for an only connected system, the limit points of any trajectory is not isolated. In the latter case, this paper also gives the strict expression of the limit points via the Bloch vector framework of density operators.