Multipartite synchronization residuals in driven-dissipative spin networks

Technical Summary: Multipartite Synchronization Residuals in Driven-Dissipative Spin Networks

Problem Statement
Quantum synchronization involves the emergence of phase-locked collective oscillations in coupled dissipative quantum systems driven out of equilibrium. Unlike classical synchronization, which is defined by trajectory convergence in phase space, the quantum regime lacks a globally valid phase-space description due to quantum fluctuations. Consequently, operational diagnostics are required, often relying on correlation functions, mutual information, or phase-space quasiprobability distributions.

A central question addressed in this work is whether phase-sensitive synchronization measures in multipartite systems adhere to additive constraints analogous to the subadditivity (SA) and strong subadditivity (SSA) inequalities that govern entropic measures (e.g., von Neumann entropy). While entropic correlations are strictly bounded by these inequalities, it is not guaranteed that synchronization measures—particularly those involving genuine multipartite phase locking—can be fully decomposed into pairwise contributions. The authors investigate whether "synchronization residuals" (the difference between collective and reduced subsystem contributions) remain non-negative or if they can become negative, signaling collective structures invisible to bipartite characterizations.

Methodology
The authors introduce a phase-space measure of quantum synchronization based on the first angular moments of phase distributions derived from Husimi-Q quasiprobability functions. This approach provides a finite-dimensional analogue to the Kuramoto order parameter.

1. Definition of Residuals:
   Inspired by information-theoretic hierarchies, the authors define synchronization residuals for bipartite and tripartite systems:

   • Bipartite Residual ($\Delta^{(2)}_R$): $R_1(t) + R_2(t) - R_{12}(t)$
   • Tripartite Residual ($\Delta^{(3)}_R$): $R_{12}(t) + R_{23}(t) - R_2(t) - R_{123}(t)$
     Here, $R$-functions represent the magnitude of the first angular moment of the phase distribution (a measure of phase localization). $R_{12}$ and $R_{123}$ are calculated from joint distributions projected onto relative phase variables, while $R_1, R_2, R_3$ are derived from marginal distributions.

2. Physical System:
   The study utilizes a driven-dissipative quantum Rabi network in the dispersive adiabatic regime.

   • Two-Qubit Case: Two spatially separated qubits, each coupled to a local cavity mode via Jaynes-Cummings interaction, with coherent hopping between cavities mediating effective qubit-qubit interaction.
   • Three-Qubit Case: Three qubits coupled to local cavities arranged in a triangular topology with mutual photon hopping.
   • Dynamics: The cavity modes are adiabatically eliminated to obtain an effective Hamiltonian. The system's evolution is solved using the Lindblad master equation, accounting for local relaxation and pure local dephasing.

3. Analysis:
   The Husimi-Q distributions ($Q$) are computed for the collective and reduced density matrices. Phase distributions are obtained via angular marginalization of these $Q$ functions. The synchronization measures ($R$) and residuals ($\Delta_R$) are calculated from these distributions. For comparison, the authors also compute the corresponding entropy-based residuals ($\Delta_S$) using von Neumann entropy to test the standard SSA inequality.

Key Results

• Two-Qubit System: In the steady state, the collective Husimi-Q distribution exhibits stronger phase localization along the azimuthal direction compared to the individual qubit distributions. The bipartite synchronization residual $\Delta^{(2)}_R$ remains non-negative throughout the evolution. This indicates that the collective phase localization is bounded by the sum of single-qubit contributions, consistent with a subadditivity-like structure in the studied parameter regime. The corresponding entropy-based residual $\Delta^{(2)}_S$ is also non-negative.
• Three-Qubit System: The three-qubit nonequilibrium steady state exhibits a negative tripartite synchronization residual ($\Delta^{(3)}_R < 0$). This violation of the SSA-like inequality implies that the collective phase synchronization cannot be fully described by a pairwise decomposition; it signifies the presence of genuine collective phase locking involving all three subsystems simultaneously.
• Entropy Comparison: In contrast to the phase-sensitive measure, the entropy-based tripartite residual $\Delta^{(3)}_S$ remains non-negative, satisfying the standard strong subadditivity inequality.

Significance and Claims
The paper claims that phase-sensitive synchronization measures and entropic correlation measures probe distinct aspects of open-system dynamics.

• The negativity of the tripartite synchronization residual demonstrates that multipartite synchronization in driven-dissipative systems constitutes a class of quantum correlation that is not reducible to pairwise interactions, analogous to genuine multipartite entanglement.
• The fact that the entropy-based residual remains non-negative while the phase-based residual becomes negative underscores that standard information-theoretic quantities (like mutual information) may fail to capture specific features of relative phase localization and locking.
• The authors conclude that multipartite synchronization in these systems represents an inequivalent class of quantum correlation, possessing features not directly reflected in conventional information-theoretic quantities. The Husimi-Q phase-space framework is presented as a useful tool for examining these effects in finite-dimensional systems.