The exact information cost of randomized measurements in quantum metrology

Abstract

Randomized measurements provide state-independent readout for multiparameter quantum sensing. Although 3-designs retain a constant fraction of the quantum Fisher information (QFI) for every pure state, we show that 2-designs need not. In dimension \(d\), complete mutually unbiased bases (MUBs) retain exactly \(1/(d+1)\) of the QFI in the worst state and direction, attaining the universal lower bound for 2-designs built from orthonormal bases; symmetric informationally complete (SIC) measurements retain exactly \((1-1/\sqrt{d+1})/d\). The loss comes from aligned blind spots. Relative to a basis, pure-state changes split into population and phase parts of equal dimension, but probabilities respond only to the former. Mutual unbiasedness turns a population change for one basis into a phase change for every other. The worst case already occurs for a qubit encoded in two levels. Independent random bases instead scatter their blind spots. In sufficiently large dimensions, five Haar-random bases retain, with high probability, a dimension-independent QFI fraction for every pure state and direction. Random Clifford measurements on any number of qubits retain exactly one third. Thus blind-spot geometry, not second-order symmetry alone, controls worst-case metrological performance.