A quantum analog of Huygen's clock: noise-induced synchronization

We propose a quantum analogue of the Huygens clock, in which the phases of two spins achieve synchronization through their interaction with a shared environment. The environment functions analogously to the escapement mechanism in a mechanical clock, regulating the gear train and permitting the advancement of timing in discrete intervals. In our proposed model, the relative phase of the two spins become synchronized through interaction with a mutual, correlated, environment. We show that for a system of qubits, several arguments can be made that significantly reduce the cardinality of the set of allowed measurements and, hence, the complexity of the problem. We present a numerically efficient method to calculate the degree of quantumness that exists in the correlations of our final density matrix. This method also provides a tight upper bound for when the system is described by rank-3 and rank-4 density matrices.