Noise-induced Quantum Synchronization and Entanglement in a Quantum Analog of Huygens' Clock

We propose a quantum analogue of the Huygens clock, where the phases of two spins synchronize through

their interaction with a shared environment. This environment acts like the escapement mechanism in a

mechanical clock, regulating the gear train and allowing discrete timing advances. In our model, the relative

phases of the two spins synchronize via a mutual correlated environment. We demonstrate that for a system of

qubits, several arguments can significantly reduce the cardinality of allowed measurements, thus simplifying

the problem. We present a numerically efficient method to calculate the degree of quantumness in the

correlations of the final density matrix, providing a tight upper bound for rank-3 and rank-4 density matrices.

We conclude by suggesting a potential realization of noise-induced synchronization between two nuclear spins

coupled to a common ancilla undergoing dynamical decoupling.