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Mathematical Methods in the Applied Sciences

Well‐posedness of quantum stochastic differential equations driven by fermion Brownian motion in noncommutative Lp‐space

Guangdong Jing, Penghui Wang, Shan Wang

Published Mar 24, 2024• 3 citations
Publisher

Abstract

This paper is concerned with quantum stochastic differential equations driven by the fermion field in noncommutative space for . First, we investigate the existence and uniqueness of ‐solutions of quantum stochastic differential equations in an infinite time horizon by using the noncommutative Burkholder–Gundy inequality given by Pisier and Xu and the noncommutative generalized Minkowski inequality. Then, we investigate the stability, self‐adjointness, and Markov properties of ‐solutions and analyze the error of numerical schemes of quantum stochastic differential equations. The results of this paper can be utilized for investigating the optimal control of quantum stochastic systems with infinite dimensions.

DOI
10.1002/mma.9953

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This paper is concerned with quantum stochastic differential equations driven by the fermion field in noncommutative space for . First, we investigate the existence and uniqueness of ‐solutions of quantum stochastic differential equations in an infinite time horizon by using the noncommutative Burkholder–Gundy inequality given by Pisier and Xu and the noncommutative generalized Minkowski inequality. Then, we investigate the stability, self‐adjointness, and Markov properties of ‐solutions and analyze the error of numerical schemes of quantum stochastic differential equations. The results of this paper can be utilized for investigating the optimal control of quantum stochastic systems with infinite dimensions.
ABSTRACT This manuscript focuses on the study of optimal quantum control problem for the quantum stochastic model in noncommutative space. The existence and uniqueness of the quantum stochastic differential equation driven by a fermion Brownian motion with bounded delay in finite time interval are established by employing the noncommutative Burkholder–Gundy inequality, generalised Minkowski inequality and generalised Banach fixed‐point theorem. The proposed system is formulated to control the quantum stochastic differential equation and to maximise the probability that the quantum state will reach the target state. By using the Pontryagin maximum principle, optimal quantum control theory is established for the proposed system. Furthermore, an application of the algorithm is presented through numerical simulations which validate the theoretical results.
This paper aims to establish second order necessary conditions for optimal control in quantum stochastic systems. We employ a variational approach, analogous to methods in classical stochastic control, to analyze systems governed by quantum stochastic differential equations driven by fermionic Brownian motion, where the control enters both the drift and diffusion terms. This result provides a theoretical foundation for further exploration of optimization problems and their practical applications in the field of quantum stochastic control.
In this paper, the Pontryagin-type maximum principle for optimal control of quantum stochastic systems in fermion fields is obtained. These systems have gained significant prominence in numerous quantum applications ranging from physical chemistry to multi-dimensional nuclear magnetic resonance experiments. Furthermore, we establish the existence and uniqueness of solutions to backward quantum stochastic differential equations driven by fermion Brownian motion. The application of noncommutative martingale inequalities and the martingale representation theorem enables this achievement.
This paper aims to establish second order necessary conditions for optimal control in quantum stochastic systems. We employ a variational approach, analogous to methods in classical stochastic control, to analyze systems governed by quantum stochastic differential equations driven by fermionic Brownian motion, where the control enters both the drift and diffusion terms. This result provides a theoretical foundation for further exploration of optimization problems and their practical applications in the field of quantum stochastic control.
Abstract. In this paper, we derive a Pontryagin-type maximum principle for optimal control of quantum systems in fermion fields. These systems have gained significant prominence in numerous quantum applications ranging from physical chemistry to multidimensional nuclear magnetic resonance experiments. Furthermore, we establish the existence and uniqueness of solutions to backward quantum stochastic differential equations driven by fermion Brownian motion. The application of noncommutative martingale inequalities and the martingale representation theorem enables this achievement.