We present a strategy for obtaining optimal probe states and measurement schemes in a class of noiseless multiparameter estimation problems with symmetry among the generators. The key to the framework is the introduction of a set of quantum metrology conditions, analogous to the quantum error correction conditions of Knill and Laflamme, which are utilized to identify probe states that saturate the multiparameter quantum Cramér-Rao bound. Similar to finding two-dimensional irreps for encoding a logical qubit in error correction, we identify trivial irreps of finite groups that guarantee the satisfaction of the quantum metrology conditions. To demonstrate our framework, we analyze the $\mathrm{SU}\left(2\right)$ estimation with symmetric states in which three parameters define a global rotation of an ensemble of $N$ qubits. For even $N$, we find that tetrahedral symmetry and, with fine-tuning, $S_3$ symmetry, are minimal symmetry groups providing optimal probe states for $\mathrm{SU}\left(2\right)$ estimation, but that the quantum metrology conditions can also be satisfied in an entanglement-assisted setting by using a maximally entangled state of two spin-$N/2$ representations for any $N$. By extending the multiparameter method of moments to noncommuting observables, we use the quantum metrology conditions to construct a measurement scheme that saturates the multiparameter quantum Cramér-Rao bound for small rotation angles.

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• Quantum error correction
• Quantum information theory
• Quantum metrology