Coherent feedback control for linear quantum systems and two-strategy evolutionary game theory

Abstract

The thesis is mainly concerned with two fields: coherent feedback control for linear quantum systems and two-strategy evolutionary game theory. Three topics are considered: 1. Design mixed linear quadratic Gaussian (LQG) and H∞ coherent feedback controllers for linear quantum systems. 2. Develop a new classical strategy model to solve the problem that defectors always dominate cooperators in a static two-strategy game. 3. Generalize the classical game theory into the quantum domain and state the advantages of quantum strategies over classical strategies. For topic 1, a class of closed-loop linear quantum systems is formulated in terms of quantum stochastic differential equations (QSDEs) in quadrature form, where both the plant and the controller are quantum systems. Under this framework, the mixed feedback control problem is synthesized. After proving a general result for the lower bound of LQG index, two methods, rank constrained LMI method and genetic-algorithm-based method, are proposed for controller design. A passive system (cavity) and a non-passive one (degenerate parametric amplifier, DPA), used as numerical examples, demonstrate the effectiveness of these two proposed algorithms. Furthermore, the superiority of genetic algorithm (GA) is verified by the comparison between the numerical results of two proposed methods. For topic 2, by adding tags to the game players, a class of classical two-strategy evolutionary model with finite population is proposed in Chapter 4. Tags represent different characteristics of individuals in the game, and each player can choose how many tags she/he expresses in the game. When an individual expresses more tags, she/he will obtain higher probability to have common tags, which will make probabilistically higher payoff. Nevertheless, the players should also pay for the expressed tags, the cost of expressed tags increases with its numbers arising. Upon this model, the evolutionary dynamics and stationary distributions of infinite time are synthesized. Finally, three kinds of feasible strategies for C-strategy individual invading into D-strategy population are obtained from numerical examples. Topic 3 is presented in Chapter 5. Upon the standard classical memory-one Iterated Prisoner's Dilemma (IPD) game, the classical zero-determinant (ZD) strategies in Press and Dyson (2012) is introduced. Then a class of two-strategy evolutionary game theory, named quantum zero-determinant (ZD) strategies, is attained by generalizing the classical ZD strategies into the quantum domain. Three kinds of numerical examples are given, which illustrate that the quantum zero-determinant strategies have significant advantage over the classical zero-determinant strategies. Meanwhile, when both players choose quantum zero-determinant strategies, it is the same as the classical case, which are named quasi-classical zero-determinant (ZD) strategies.