Optimal Multi-Parameter Metrology: Vector Field Sensing with Two Qubits

We solve exactly the problem of quantum-limited three-dimensional magnetometry with two qubits, the smallest possible system for simultaneous estimation of three parameters. This minimal setting intensifies measurement incompatibility, requiring analysis via the Holevo Cramér-Rao bound (HCRB). Using and extending an asymptotic Bayesian optimization method [1], we identify optimal states and measurements that realize a novel multiparameter sensor—the quantum compass—which attains the minimal HCRB locally while remaining robust over a broad parameter range. We further report an experimental proof-of-principle demonstration [2] of this smallest optimal vector sensor using a pair of trapped ions. The experiment confirms that the sensor reaches the ultimate HCRB precision and outperforms even an ideal implementation of the best-known adaptive classical strategy by 1.623(4) dB in single-shot estimation, this performance is maintained over a broad range of unambiguous estimation. Our results establish the quantum compass as both a theoretical benchmark and an experimentally viable testbed for exploring incompatibility tradeoffs and advancing multiparameter quantum sensing.