Consensus And Information Redistribution In Quantum Networks

Abstract

Consensus algorithms are mechanisms driving nodes of a network to agreement. In quantum networks they also redistribute information encoded in the global quantum state, allowing global properties to be inferred from local measurements without reconstructing the global state. Motivated by this, we investigate decentralized quantum consensus in qudit networks governed by discrete random unitary dynamics. We characterize the analytic asymptotic dynamics, identify global information preserved by the evolution, and derive conditions for which different notions of quantum consensus emerge. A general theorem is derived that establishes a unified framework for extracting globally encoded information from local measurements beyond consensus states. First, we analyze CNOT networks and determine which global information can be recovered by individual nodes and which requires joint measurements on cooperating nodes. The CNOT model is well understood and presents a case with entanglement creation - a characteristic quantum feature of networks worth analyzing in detail. Second, we identify the complete class of two-qubit interactions that asymptotically achieve consensus on the average number of excitations, thereby establishing a quantum analogue of classical average consensus. We further show that joint measurements on k cooperating nodes recover exactly the first k moments of the excitation-number distribution.

The paper asks which global properties of a quantum-network state survive decentralized dynamics and become accessible through measurements on individual nodes or cooperating subsets. It considers discrete random-unitary channels $\mathcal P(\rho)=\sum_e p_eU_e\rho U_e^\dagger$, with pairwise interactions assigned to network edges. Using the peripheral eigenspaces of these channels, it derives necessary and sufficient algebraic conditions for expectation-value, reduced-state, and permutation-symmetric consensus. A second theorem characterizes stationary global information accessible through a chosen local observable: its projection onto the stationary attractor space identifies the corresponding conserved global observable, provided its overlaps with nonstationary attractors vanish. The construction extends to multi-node measurements and, as stated in the paper, discrete unital Markov channels. Two applications illustrate the framework. Strongly connected CNOT networks with $n>2$ have a five-dimensional stationary attractor space and reach permutation-symmetric consensus. Individual nodes can recover ground-state and uniform-superposition probabilities and an imaginary coherence component, whereas other preserved quantities require two-node measurements. For excitation-number consensus, the paper classifies two-qubit interactions by analyzing the two-node attractor equations, then gives an $n+3$-dimensional attractor structure for the selected mechanisms at $n>2$. Proper subsets of $m$ nodes access exactly the first $m$ moments of the conserved excitation-number distribution; explicit moment observables are constructed recursively. The analysis concerns asymptotic accessibility. Observable reconstruction becomes harder for large attractor spaces. Non-unital dynamics and a fuller characterization of recoverable properties, particularly in entangled consensus states, are left for future work.