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Adaptive techniques have important potential for wide applications inenhancing precision of quantum parameter estimation. We present a recursivelyadaptive quantum state tomography (RAQST) protocol for finite dimensionalquantum systems and experimentally implement the adaptive tomography protocolon two-qubit systems. In this RAQST protocol, an adaptive measurement strategyand a recursive linear regression estimation algorithm are performed. Numericalresults show that our RAQST protocol can outperform the tomography protocolsusing mutually unbiased bases (MUB) and the two-stage MUB adaptive strategyeven with the simplest product measurements. When nonlocal measurements areavailable, our RAQST can beat the Gill-Massar bound for a wide range of quantumstates with a modest number of copies. We use only the simplest productmeasurements to implement two-qubit tomography experiments. In the experiments,we use error-compensation techniques to tackle systematic error due tomisalignments and imperfection of wave plates, and achieve about 100-foldreduction of the systematic error. The experimental results demonstrate thatthe improvement of RAQST over nonadaptive tomography is significant for stateswith a high level of purity. Our results also show that this recursivelyadaptive tomography method is particularly effective for the reconstruction ofmaximally entangled states, which are important resources in quantuminformation.

Recursively Adaptive Quantum State Tomography: Theory and Two-qubit Experiment