Quantum error correction-inspired multiparameter quantum metrology

We present a novel 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 Cram\'{e}r-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 SU(2) 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 SU(2) 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 non-commuting observables, we use the quantum metrology conditions to construct a measurement scheme that saturates the multiparameter quantum Cram\'{e}r-Rao bound for small rotation angles.