Multiparameter quantum metrology for general and adaptive strategies

Abstract

We develop a conic-programming framework for multiparameter estimation of quantum processes under different classes of sensing strategies. Estimation precision depends on the choice of input probe state and measurement and, when a channel is used multiple times, on the arrangement of channel uses and intermediate controls. We specify different strategy classes through linear normalization constraints on generalized measurements, and formulate the local estimation problem using linear constraints and a linear objective function. We show that the separable-cone program gives the exact precision limit under the local estimation conditions considered here, while cone relaxations yield Cramér-Rao-type lower bounds. For $N$ uses of an identical quantum channel, we prove an exact $1/N$ scaling law for an RLD-based positive-cone lower bound, even for general process-matrix testers that allow indefinite causal order. Under the stated conditions on the channel and its parameter dependence, the minimum weighted mean-square error also scales as $1/N$. This extends an RLD-based restriction on Heisenberg scaling from adaptive strategies to this broader class. For fixed-order adaptive strategies, we show that each process-side conic bound equals the corresponding state-estimation bound optimized over sensor combs. We establish this equality by constructing a sensor comb from the process-side normalization operator. A noisy three-parameter field-sensing model illustrates the framework.