Near-optimal pure state estimation with adaptive Fisher-symmetric measurements

Abstract

Quantum state estimation is important for various quantum information processes, including quantum communications, computation, and metrology, which require the characterization of quantum states for evaluation and optimization. We present a three-stage adaptive method for estimating arbitrary $d$-dimensional pure quantum states using locally informationally complete Fisher symmetric measurements (FSM) and a single-shot measurement basis. We derive finite-sample high-probability error bounds for the protocol and demonstrate that our approach scales as $O(d/N)$ for large sample sizes, thereby guaranteeing the advantage of adaptation. Moreover, numerical simulations indicate that the protocol achieves an average infidelity close to the optimal given by the Gill-Massar lower bound (GMB). The total number of measurement outcomes scales linearly with $7d-3$, avoiding the need for collective measurements on multiple copies of the unknown state. This work highlights the potential of adaptive estimation techniques in quantum state characterization while maintaining efficiency in the number of measurement outcomes.