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We study quantum-limited 3D magnetometry using two qubits. Two qubits formthe smallest multi-qubit system for 3D magnetometry, the simultaneousestimation of three phases, as it is impossible with a single qubit. We providean analytical expression for the Holevo-Cram\'er-Rao bound (HCRB),thefundamental attainable quantum bound of multiparameter estimation, for 3Dmagnetometry using two-qubit pure states and show its attainability by rank-1projective measurements. We also examine the attainability of the HCRB in thepresence of dephasing noise using numerical methods. While attaining the HCRBmay require collective measurements over infinitely many copies, we find thatfor high noise the HCRB is practically saturated by two copies only. In the lownoise regime, up to three copies are unable to attain the HCRB. More generally,we introduce new multiparameter channel bounds to compare quantum-classical andclassical-quantum strategies where multiple independent copies of the state areentangled before or after recording the parameters respectively. We find thattheir relative performance depends on the noise strength, withtheclassical-quantum strategy performing better for high noise. We end withshallow quantum circuits that approach the fundamental quantum limit set by theHCRB for two-qubit 3D magnetometry using up to three copies.

Attainability of the Holevo-Cram\'er-Rao bound for two-qubit 3D magnetometry