Adaptive quantum state tomography via linear regression estimation: Theory and two-qubit experiment

Abstract

Adaptive techniques have great potential for wide application in enhancing the precision of quantum parameter estimation. We present an adaptive quantum state tomography protocol for finite dimensional quantum systems and experimentally implement the adaptive tomography protocol on two-qubit systems. In this adaptive quantum state tomography protocol, an adaptive measurement strategy and a recursive linear regression estimation algorithm are performed. Numerical results show that our adaptive quantum state tomography protocol can outperform tomography protocols using mutually unbiased bases and the twostage mutually unbiased bases adaptive strategy, even with the simplest product measurements. When nonlocal measurements are available, our adaptive quantum state tomography can beat the Gill-Massar bound for a wide range of quantum states with a modest number of copies. We use only the simplest product measurements to implement two-qubit tomography experiments. In the experiments, we use error-compensation techniques to tackle systematic error due to misalignments and imperfection of wave plates, and achieve about a 100-fold reduction of the systematic error. The experimental results demonstrate that the improvement of adaptive quantum state tomography over nonadaptive tomography is significant for states with a high level of purity. Our results also show that this adaptive tomography method is particularly effective for the reconstruction of maximally entangled states, which are important resources in quantum information.

Practical adaptive quantum tomography

“…A generic approach using the maximum likelihood estimator and measurements minimizing the expected variance also showed an improvement over standard quantum tomography [21]. This has been made more practical through use of a recursive least-squares formula in Qi et al [25]. Below we will see that our choice of heuristic for adaptation may lead the least squares estimator to fail due to ill-conditionedness.…”

“…PAQT intelligently selects new measurements based on the outcomes of previous ones [10,[17][18][19][20][21]. Adaptivity has been experimentally demonstrated [14,15,[22][23][24][25], but is not currently standard practice. Though adaptivity increases accuracy, the computational costs incurred outweigh that of simply repeating standard measurements many times.…”

Quantum tomography by regularized linear regressions

“…Linear regression was also applied to quantum tomography with incomplete measurement bases for low-rank, e.g., Alquier et al (2013); Gross (2011); Gross et al (2010) or sparse, e.g., Cai et al (2016) quantum states in view of the insights from compressed sensing, where a small number of measurement bases was proven to be enough for the recovery of a high-dimensional quantum state with high probability as the dimension increases. Recently, linear regression method was also generalized to the adaptive measurement case where selection of the measurement basis depends on the previous measurement outcomes (Qi et al, 2017).…”

Benchmarking Maximum-Likelihood State Estimation with an Entangled Two-Cavity State

“…If the initial quantum state is known, the method can also be efficiently implemented for the parameter estimation [36]. Finally, the analysis of the structure of the effect matrices provides a guide for designing adaptive reconstruction procedures [37][38][39] by selecting optimal measurements. We acknoweldge support by European Research Council (DECLIC and TRENSCRYBE projects), by European Community (SIQS project) and by the Agence Nationale de la Recherche (QuDICE project).…”