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Reduced-State Synchronization of Quantum Networks: Convergence, Graphical Information Hierarchy, and the Missing Symmetry

Guodong Shi, Shuangshuang Fu, Ian R. Petersen

Published Oct 22, 2014• 4 citations
OpenAlex

Abstract

We establish a thorough treatment of reduced-state synchronization for qubit networks with the aim of driving the qubits ’ reduced states to a common trajectory. The evolution of the quantum network’s state is described by a master equation, where the network Hamil-tonian is either a direct sum or a tensor product of identical qubit Hamiltonians, and the coupling terms are given by a set of permutation operators over the network. The per-mutations introduce naturally quantum directed interactions. We show that reduced-state synchronization is achieved if and only if the quantum interaction graphs corresponding to the permutation operators form a strongly connected union graph. The proof is based on an algebraic analysis making use of the Perron-Frobenius theorem for non-negative matri-ces. The convergence rate and the limiting orbit are explicitly characterized. Numerical examples are provided illustrating the obtained results. Further, we investigate the miss-ing symmetry in the reduced-state synchronization from a graphical point of view. The information-flow hierarchy in quantum permutation operators is characterized by different layers of information-induced graphs. We show that the quantum synchronization equation is by nature equivalent to several parallel cut-balanced consensus processes, and a neces-sary and sufficient condition is obtained for quantum reduced-state synchronization under switching interactions applying recent work of Hendrickx and Tsitsiklis.

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We establish a thorough treatment of reduced-state synchronization for qubit networks with the aim of driving the qubits ’ reduced states to a common trajectory. The evolution of the quantum network’s state is described by a master equation, where the network Hamil-tonian is either a direct sum or a tensor product of identical qubit Hamiltonians, and the coupling terms are given by a set of permutation operators over the network. The per-mutations introduce naturally quantum directed interactions. We show that reduced-state synchronization is achieved if and only if the quantum interaction graphs corresponding to the permutation operators form a strongly connected union graph. The proof is based on an algebraic analysis making use of the Perron-Frobenius theorem for non-negative matri-ces. The convergence rate and the limiting orbit are explicitly characterized. Numerical examples are provided illustrating the obtained results. Further, we investigate the miss-ing symmetry in the reduced-state synchronization from a graphical point of view. The information-flow hierarchy in quantum permutation operators is characterized by different layers of information-induced graphs. We show that the quantum synchronization equation is by nature equivalent to several parallel cut-balanced consensus processes, and a neces-sary and sufficient condition is obtained for quantum reduced-state synchronization under switching interactions applying recent work of Hendrickx and Tsitsiklis.
We consider reduced-state synchronization of qubit networks with the aim of driving the qubits' reduced states to a common trajectory. The evolution of the quantum network's state is described by a master equation, where the network Hamiltonian is either a direct sum or a tensor product of identical qubit Hamiltonians, and the coupling terms are given by a set of permutation operators over the network. The permutations introduce naturally quantum directed interactions. Part I of the paper establishes synchronization conditions for fixed quantum interactions. In this part of the paper, we further investigate the missing symmetry in the reduced-state synchronization from a graphical point of view. The information-flow hierarchy in quantum permutation operators is characterized by different layers of information-induced graphs, based on which a clear bridge between quantum and classical consensus dynamics is built. We show that the quantum synchronization equation is by nature equivalent to a cut-balanced consensus process. Then a necessary and sufficient condition is obtained for reaching quantum reduced-state synchronization in light of recent work by Hendrickx and Tsitsiklis [19].
We consider reduced-state synchronization of qubit networks with the aim of driving the qubits' reduced states to a common trajectory. The evolution of the quantum network's state is described by a master equation, where the network Hamiltonian is either a direct sum or a tensor product of identical qubit Hamiltonians, and the coupling terms are given by a set of permutation operators over the network. The permutations introduce naturally quantum directed interactions. This part of the paper focuses on convergence conditions. We show that reduced-state synchronization is achieved if and only if the quantum permutations form a strongly connected union graph. The proof is based on an algebraic analysis making use of the Perron-Frobenius theorem for non-negative matrices. The convergence rate and the limiting orbit are explicitly characterized. Numerical examples are provided illustrating the obtained results.
In this paper, we study consensus seeking of quantum networks under directed interactions defined by a set of permutation operators among a network of qubits. The state evolution of the quantum network is described by a continuous-time master equation, for which we establish an unconditional convergence result indicating that the network state always converges with the limit determined by the generating subgroup of the permutations making use of the Perron-Frobenius theory. We also give a tight graphical criterion regarding when such limit admits a reduced-state consensus. Further, we provide a clear description to the missing symmetry in the reduced-state consensus from a graphical point of view, where the information-flow hierarchy in quantum permutation operators is characterized by different layers of information-induced graphs. Finally, we investigate quantum synchronization in the presence of network Hamiltonian, study quantum consensus conditions under switching interactions, and present a few numerical examples illustrating the obtained results.
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.
We study consensus seeking of quantum networks under directed balanced interactions defined by a set of permutation operators among a network of qubits. The state evolution of the quantum network is described by a continuous-time master equation. We show that the network state always converges along the equation, and the convergence limit is determined by the generating subgroup of the permutation operators. We also give a tight graphical criterion regarding when such limit admits a reduced-state consensus. Quantum consensus conditions under switching interactions are also established with the help of cut-balanced consensus processes in the classical setting.
In this paper, we propose and study a master-equation based approach to drive a quantum network with n qubits to a consensus (symmetric) state introduced by Mazzarella et al. The state evolution of the quantum network is described by a Lindblad master equation with the Lindblad terms generated by continuous-time swapping operators, which also introduce an underlying interaction graph. We establish a graphical method that bridges the proposed quantum consensus scheme and classical consensus dynamics by studying an induced graph (with 22nnodes) of the quantum interaction graph (with n qubits). A fundamental connection is then shown that quantum consensus over the quantum graph is equivalent to componentwise classical consensus over the induced graph, which allows various existing works on classical consensus to be applicable to the quantum setting. Some basic scaling and structural properties of the quantum induced graph are established via combinatorial analysis. Necessary and sufficient conditions for exponential and asymptotic quantum consensus are obtained, respectively, for switching quantum interaction graphs. As a quantum analogue of classical synchronization of coupled oscillators, quantum synchronization conditions are also presented, in which the reduced states of all qubits tend to a common trajectory.