Dynamic Programming Principle and Stabilization for Mean-Field Quantum Filtering Systems

TL;DR

This paper addresses the challenge of designing feedback controls for continuously monitored quantum systems in infinite dimensions.It develops a rigorous dynamic programming framework by embedding density operators into the Hilbert-Schmidt space $\mathcal{B}_2(\mathbb{H})$, deriving Fréchet differentiability and an $HJB$ equation for the value function within the quantum filtering setting.In the concrete setting of $N$ Ising-coupled qubits, it shows quantum state reduction and exponential stabilization under continuous-time feedback, and analyzes the mean-field limit as $N \to \infty$ to obtain a tractable surrogate dynamics.It also formulates a quantum differential game with two players and analyzes both cooperative and competitive scenarios, demonstrating local feedback based on reduced states and establishing well-posedness via fixed-point arguments.

Abstract

Working within the quantum filtering framework, we establish a dynamic programming principle in an infinite-dimensional setting by embedding the state space into the Hilbert-Schmidt space. We then study a stabilization problem for continuously monitored Ising-coupled qubits and, in the mean-field limit, demonstrate quantum state reduction together with exponential convergence toward prescribed eigenstates under suitable feedback laws.

Figures (5)

• Figure 1: Quantum state reduction: convergence of ${\gamma}_{t}$ toward $\{\rho_g,\rho_e\}$.
• Figure 2: Stabilization of ${\gamma}_{t}$ toward ${\rho_{g}}$ when the control $\alpha^{\star}$ is applied.
• Figure 3: Two qubits interact via an Ising Hamiltonian $\sigma_z\otimes\sigma_z$. Alice and Bob each perform a continuous $z$-measurement and apply local feedback along the $x$-axis.
• Figure 4: In one realization, quantum state reduction: convergence of $\boldsymbol{\rho}_{t}$ toward $\textcolor{blue}{\rho_{eg}}$
• Figure 5: In one realization, stabilization of $\boldsymbol{\rho}_{t}$ toward $\textcolor{red}{\rho_{ge}}$, when each player adopt control strategy $(\alpha_1,\alpha_2)$

Theorems & Definitions (11)

• Definition 1: Differentiability
• Lemma 1
• proof
• Theorem 1
• proof
• Remark 1
• Remark 2
• Theorem 2
• proof
• Remark 3