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Derivation of the Lindblad Equation for Open Quantum Systems and Its Application to Mathematical Modeling of the Process of Decision Making

Xingdong Zuo

Published Jan 1, 2014• 0 citations
OpenAlex

Abstract

In the theory of open quantum systems, a quantum Markovian master equation, the Lindblad equation, reveals the most general form for the generators of a quantum dynamical semigroup. In this thesis, we present the derivation of the Lindblad equation and several examples of Lindblad equations with their analytic and numerical solutions. The graphs of the numerical solutions illuminate the dynamics and the stabilization as time increases. The corresponding von Neumann entropies are also presented as graphs. Moreover, to illustrate the difference between the dynamics of open and isolated systems, we prove two theorems about the conditions for stabilization of the solutions of the von Neumann equation which describes the dynamics of the density matrix of open quantum systems. It shows that the von Neumann equation is not satisfied for modelling dynamics in the cognitive contextin general. Instead, we use the Lindblad equation to model the mental dynamics of the players in the game of the 2-player prisoner’s dilemma to explain the irrational behaviors of the players. The stabilizing solution will lead the mental dynamics to an equilibrium state, which is regarded as the termination of the comparison process for a decision maker. The resulting pure strategy is selected probabilistically by performing a quantum measurement. We also discuss two important concepts, quantum decoherence and quantum Darwinism. Finally, we mention a classical Neural Network Master Equation introduced by Cowan and plan our further works on an analogous version for the quantum neural network by using the Lindblad equation.

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In the theory of open quantum systems, a quantum Markovian master equation, the Lindblad equation, reveals the most general form for the generators of a quantum dynamical semigroup. In this thesis, we present the derivation of the Lindblad equation and several examples of Lindblad equations with their analytic and numerical solutions. The graphs of the numerical solutions illuminate the dynamics and the stabilization as time increases. The corresponding von Neumann entropies are also presented as graphs. Moreover, to illustrate the difference between the dynamics of open and isolated systems, we prove two theorems about the conditions for stabilization of the solutions of the von Neumann equation which describes the dynamics of the density matrix of open quantum systems. It shows that the von Neumann equation is not satisfied for modelling dynamics in the cognitive contextin general. Instead, we use the Lindblad equation to model the mental dynamics of the players in the game of the 2-player prisoner’s dilemma to explain the irrational behaviors of the players. The stabilizing solution will lead the mental dynamics to an equilibrium state, which is regarded as the termination of the comparison process for a decision maker. The resulting pure strategy is selected probabilistically by performing a quantum measurement. We also discuss two important concepts, quantum decoherence and quantum Darwinism. Finally, we mention a classical Neural Network Master Equation introduced by Cowan and plan our further works on an analogous version for the quantum neural network by using the Lindblad equation.
We proceed towards an application of the mathematical formalism of quantum mechanics to cognitive psychology — the problem of decision-making in games of the Prisoners Dilemma type. These games were used as tests of rationality of players. Experiments performed in cognitive psychology by Shafir and Tversky [1, 2], Croson [3], Hofstader [4, 5] demonstrated that in general real players do not use "rational strategy" provided by classical game theory; this psychological phenomenon was called the disjunction effect. We elaborate a model of quantum-like decision making which can explain this effect ("irrationality" of plays). Our model is based on quantum information theory. The main result of this paper is the derivation of Gorini-Kossakowski-Sudarshan-Lindblad equation whose equilibrium solution gives the quantum state used for decision making. It is the first application of this equation in cognitive psychology.
We propose a quantum-mechanical framework to describe the dynamics of human decision making, where the interactions between the open quantum system of decision makers and its surrounding environment at given temperature are relevantly taken into account. The temporal evolution of quantum system is described in terms of the time-dependent density matrix, from which we can evaluate those thermodynamic functions such as internal energy, (von Neumann) entropy and (Helmholtz) free energy. In this study we rely on the symmetrical quasi-classical (SQC) windowing approach to numerically solve the non-adiabatic quantum dynamics. We then consider the Hamiltonians to formulate the Prisoner’s Dilemma (PD) problem and calculate the population dynamics of the quantum states of two players’ decisions, where some quantum coherence or interference effects are observed. The evaluated free energy tends to decrease toward the thermodynamic equilibrium value, thus driving the dynamics of quantum system between the Nash equilibrium and Pareto optimum states. The calculated entropy often shows a tendency of temporal decrease, thus indicating the emergence of order in the quantum system. We have also applied the present scheme to the issue of the violation of the sure thing principle and found its occurrence through explicit numerical simulations for the PD problem. The present study may provide a theoretical basis to connect the dynamics of quantum decision making to the free energy principle in cognitive science.
We present a quantum‐like model of decision making in games of the Prisoner’s Dilemma type. By this model the brain processes information by using representation of mental states in complex Hilbert space. Driven by the master equation the mental state of a player, say Alice, approaches an equilibrium point in the space of density matrices. By using this equilibrium point Alice determines her mixed (i.e., probabilistic) strategy with respect to Bob. Thus our model is a model of thinking through decoherence of initially pure mental state. Decoherence is induced by interaction with memory and external environment. In this paper we study (numerically) dynamics of quantum entropy of Alice’s state in the process of decision making. Our analysis demonstrates that this dynamics depends nontrivially on the initial state of Alice’s mind on her own actions and her prediction state (for possible actions of Bob.)
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.
This paper starts with surveying the evolution of quantum-like models of cognition and decision making, transitioning from static kinematic representations to a robust dynamical framework based on open quantum systems. We provide a comprehensive analysis of the Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) master equation's application in cognitive psychology and decision making, illustrating how it models mental state evolution as a dissipative process influenced by an informational environment. We categorize dynamical regimes into Passive and Active Hamiltonians, demonstrating how non-commutation with projections on decision basis serves as a mathematical signature of cognitive agency and Quantum Escape from classical equilibria. The utility of this framework is further explored through its ability to stabilize non-Nash outcomes in strategic games, such as the Prisoner's Dilemma. Building upon this dynamical foundation, we identify ``cognitive beats'' as a signature of the internal struggle between competing ``flows of mind'' deliberated at approximately equal frequencies. Distinct from the damped oscillations of simple interference, these beats emerge from a structural tension between Liouvillian channels that generates a secondary, slow-scale modulation of conviction. This beat envelope dictates the timing of peak readiness and hesitation, providing a mathematical map of the transition between conflicting cognitive states. By resolving these nested time scales, we provide a new spectral diagnostic for the depth of cognitive agency and the complexity of the underlying deliberation process. This paper develops a theoretical framework linking GKSL dynamics with quantum-like cognition and decision-making (QCDM), highlighting how dissipative quantum models can capture features of human thought and decision processes.
We present a quantum‐like model of decision making in games of the Prisoner’s Dilemma type. By this model the brain processes information by using representation of mental states in complex Hilbert space. Driven by the master equation the mental state of a player, say Alice, approaches an equilibrium point in the space of density matrices. By using this equilibrium point Alice determines her mixed (i.e., probabilistic) strategy with respect to Bob. Thus our model is a model of thinking through decoherence of initially pure mental state. Decoherence is induced by interaction with memory and external environment. In this paper we study (numerically) dynamics of quantum entropy of Alice’s state in the process of decision making. Our analysis demonstrates that this dynamics depends nontrivially on the initial state of Alice’s mind on her own actions and her prediction state (for possible actions of Bob.)