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Journal of Physics A Mathematical and Theoretical

Quantum synchronization over quantum networks

Max A. Lohe

Published Oct 27, 2010• 142 citations
Publisher

Abstract

We consider a network of quantum oscillators in which quantum states are distributed among connected nodes by means of unitary transformations. The distributed states interact with each local state according to a time-dependent interaction Hamiltonian, which is modeled by a Hermitian operator constructed in terms of the states themselves, thereby introducing nonlinear network interactions. For qubit nodes of differing natural frequencies, we show numerically that for a sufficiently large coupling constant, synchronization of quantum nodes occurs in which the spins of all qubits are mutually aligned with a common frequency of oscillation, following initial transient configurations. We discuss the significance of quantum synchronization as a means to create copies of unknown quantum states.

DOI
10.1088/1751-8113/43/46/465301

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We consider a network of quantum oscillators in which quantum states are distributed among connected nodes by means of unitary transformations. The distributed states interact with each local state according to a time-dependent interaction Hamiltonian, which is modeled by a Hermitian operator constructed in terms of the states themselves, thereby introducing nonlinear network interactions. For qubit nodes of differing natural frequencies, we show numerically that for a sufficiently large coupling constant, synchronization of quantum nodes occurs in which the spins of all qubits are mutually aligned with a common frequency of oscillation, following initial transient configurations. We discuss the significance of quantum synchronization as a means to create copies of unknown quantum states.
Abstract We investigate a non-Abelian generalization of the Kuramoto model proposed by Lohe and given by N quantum oscillators (‘nodes’) connected by a quantum network where the wavefunction at each node is distributed over quantum channels to all other connected nodes. It leads to a system of Schrödinger equations coupled by nonlinear self-interacting potentials given by their correlations. We give a complete picture of synchronization results, given on the relative size of the natural frequency and the coupling constant, for two non-identical oscillators and show complete phase synchronization for arbitrary N > 2 identical oscillators. Our results are mainly based on the analysis of the ODE system satisfied by the correlations and on the introduction of a quantum order parameter, which is analogous to the one defined by Kuramoto in the classical model. As a consequence of the previous results, we obtain the synchronization of the probability and the current densities defined via the Madelung transformations.
Complex systems consisting of vector or matrix oscillators can synchronize to a common state characterized by a frequency matrix with distinct eigenvalues, leading to multiple frequencies of synchronization. In quantum networked systems the synchronized state is a linear combination of states corresponding to different energy levels. Suitable symmetry-breaking network interactions, however, allow only one or more such frequencies to appear. A specific example in three dimensions, where all trajectories lie on the 2-sphere, is a model of interacting spin-1 quantum angular momentum states, where synchronization to a nontrivial frequency occurs despite the presence of zero-frequency modes of oscillation.
We investigate the quantum synchronization phenomenon of the complex network constituted by coupled optomechanical systems and prove that the unknown identical quantum states can be shared or distributed in the quantum network even though the topology is varying. Considering a channel constructed by quantum correlation, we show that quantum synchronization can sustain and maintain high levels in Markovian dissipation for a long time. We also analyze the state-sharing process between two typical complex networks, and the results predict that linked nodes can be directly synchronized, but the whole network will be synchronized only if some specific synchronization conditions are satisfied. Furthermore, we give the synchronization conditions analytically through analyzing network dynamics. This proposal paves the way for studying multi-interaction synchronization and achieving effective quantum information processing in a complex network.
The phenomenon of spontaneous synchronization arises in a broad range of systems when the mutual interaction strength among components overcomes the effect of detuning. Recently it has been studied also in the quantum regime with a variety of approaches and in different dynamical contexts. We review here transient synchronization arising during the relaxation of open quantum systems, describing the common enabling mechanism in presence of either local or global dissipation. We address both networks of harmonic oscillators and spins and compare different synchronization measures.
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.
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].