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Blended Dynamics and Emergence in Open Quantum Networks

Qinghao Wen, Zihao Ren, Lei Wang, Hyungbo Shim, Guodong Shi

TL;DR

"The Blended Dynamics and Emergence in Open Quantum Networks" addresses how open quantum networks with diffusive couplings exhibit emergent behavior. It extends classical blended dynamics to quantum settings by (i) defining blended reduced-state dynamics for separable Hamiltonian and dissipation and (ii) formulating blended coherent dynamics on the induced graph for inseparable cases, with explicit coupling-gain thresholds $K_c^*$. The main contributions are rigorous results showing convergence to a common equilibrium or common trajectory in the reduced-state case, and orbit attraction to a permutation-invariant subspace in the coherent case, all accompanied by finite-time closeness guarantees to the blended dynamics. The framework provides a principled, scalable tool to analyze and design emergent classical and quantum behaviors in diffusive open quantum networks, with numerical validations illustrating the theory.

Abstract

In this paper, we develop a blended dynamics framework for open quantum networks with diffusive couplings. The network consists of qubits interconnected through Hamiltonian couplings, environmental dissipation, and consensus-like diffusive interactions. Such networks commonly arise in spontaneous emission processes and non-Hermitian quantum computing, and their evolution follows a Lindblad master equation. Blended dynamics theory is well established in the classical setting as a tool for analyzing emergent behaviors in heterogeneous networks with diffusive couplings. Its key insight is to blend the local dynamics rather than the trajectories of individual nodes. Perturbation analysis then shows that, under sufficiently strong coupling, all node trajectories tend to stay close to those of the blended system over time. We first show that this theory extends naturally to the reduced-state dynamics of quantum networks, revealing classical-like clustering phenomena in which qubits converge to a shared equilibrium or a common trajectory determined by the quantum blended reduced-state dynamics. We then extend the analysis to qubit coherent states using quantum Laplacians and induced graphs, proving orbit attraction of the network density operator toward the quantum blended coherent dynamics, establishing the emergence of intrinsically quantum and dynamically clustering behaviors. Finally, numerical examples validate the theoretical results.

Blended Dynamics and Emergence in Open Quantum Networks

TL;DR

"The Blended Dynamics and Emergence in Open Quantum Networks" addresses how open quantum networks with diffusive couplings exhibit emergent behavior. It extends classical blended dynamics to quantum settings by (i) defining blended reduced-state dynamics for separable Hamiltonian and dissipation and (ii) formulating blended coherent dynamics on the induced graph for inseparable cases, with explicit coupling-gain thresholds . The main contributions are rigorous results showing convergence to a common equilibrium or common trajectory in the reduced-state case, and orbit attraction to a permutation-invariant subspace in the coherent case, all accompanied by finite-time closeness guarantees to the blended dynamics. The framework provides a principled, scalable tool to analyze and design emergent classical and quantum behaviors in diffusive open quantum networks, with numerical validations illustrating the theory.

Abstract

In this paper, we develop a blended dynamics framework for open quantum networks with diffusive couplings. The network consists of qubits interconnected through Hamiltonian couplings, environmental dissipation, and consensus-like diffusive interactions. Such networks commonly arise in spontaneous emission processes and non-Hermitian quantum computing, and their evolution follows a Lindblad master equation. Blended dynamics theory is well established in the classical setting as a tool for analyzing emergent behaviors in heterogeneous networks with diffusive couplings. Its key insight is to blend the local dynamics rather than the trajectories of individual nodes. Perturbation analysis then shows that, under sufficiently strong coupling, all node trajectories tend to stay close to those of the blended system over time. We first show that this theory extends naturally to the reduced-state dynamics of quantum networks, revealing classical-like clustering phenomena in which qubits converge to a shared equilibrium or a common trajectory determined by the quantum blended reduced-state dynamics. We then extend the analysis to qubit coherent states using quantum Laplacians and induced graphs, proving orbit attraction of the network density operator toward the quantum blended coherent dynamics, establishing the emergence of intrinsically quantum and dynamically clustering behaviors. Finally, numerical examples validate the theoretical results.
Paper Structure (20 sections, 4 theorems, 77 equations, 7 figures)

This paper contains 20 sections, 4 theorems, 77 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Open Quantum Systems
  4. Graph Theory Essentials
  5. Blended Dynamic Theory for Classical Networks
  6. Notations
  7. Diffusive and Open Quantum Networks
  8. Emergence of Classical Behaviors
  9. Blended Reduced-state Dynamics
  10. Clustering toward a Common Equilibrium
  11. Converging to a Common Trajectory
  12. Emergence of Coherent Behaviors
  13. The Induced Graph
  14. Blended Coherent Dynamics
  15. Coherent Orbit Attraction
  16. ...and 5 more sections

Key Result

Theorem 1

Let Assumptions ass:sep hold. Assume that the blended reduced-state dynamics eq:blen1 is relaxing. Then for any given $\eta>0$, there exists some $K_c^\ast>0$ such that for any $K_c\geq K_c^\ast$, it holds along eq:obj that where $C =(1+\sqrt{2})\left( \sqrt{\sum_{j \in \mathbb{V}} \|{\bm\rho}_j(0) - \bar{{\bm\rho}}(0)\|_{\rm F}^2} + L_c\right)$ for some $L_c>0$, $\bar{{\bm\rho}}(0)=\frac{1}{n}\s

Figures (7)

  • Figure 1: The time evolution of the Frobenius distance between the reduced states ${\bm\rho}_1(t)$, ${\bm\rho}_2(t)$, and ${\bm\rho}_3(t)$ in \ref{['eq:sim1']} and the steady state ${\bm\rho}_{\rm r}$ in \ref{['eq:blen_sim1']} with varying diffusive coupling gain $K_c$ for given $\eta$.
  • Figure 2: The time evolution of Frobenius distance between the reduced states ${\bm\rho}_1(t)$, ${\bm\rho}_2(t)$, and ${\bm\rho}_3(t)$ in \ref{['eq:sim2']} and the density operator ${\bm\rho}_{\rm b}(t)$ in \ref{['eq:blen_sim2']} with varying diffusive coupling gain $K_c$ for given $T_2$ and $\eta=0.01$.
  • Figure 3: The evolutions of components of Bloch vectors of ${\bm\rho}_1(t),{\bm\rho}_2(t),{\bm\rho}_3(t)$ in \ref{['eq:sim2']} and ${\bm\rho}_{\rm b}(t)$ in \ref{['eq:blen_sim2']} with $K_c=40$. The red dots and the green dots denote the initial states and final states of the trajectories, respectively.
  • Figure 4: The illustration of the induced graph for a 3-qubit connected network governed by \ref{['eq:obj_in']}. In (a), the dashed lines represent the coupling between the induced graph nodes, arising from the term $\mathsf{A}_{\mathrm{d}} \tilde{\mathbf{y}}$ (part), while the solid lines represent the diffusive coupling due to the term $\mathsf{Q}_{\mathrm{d}} \tilde{\mathbf{y}}$.
  • Figure 5: The illustration of the blended coherent dynamics: once the nodes within each connected component have reached consensus, the coupling $\mathsf{A}_{\mathrm{d}} \tilde{\mathbf{y}}$ can be established between the connected components.
  • ...and 2 more figures

Theorems & Definitions (6)

  • Definition 1: Blended Reduced-state Dynamics
  • Theorem 1
  • Theorem 2
  • Definition 2: Blended Coherent Dynamics
  • Theorem 3
  • Theorem 4