Adaptive quantum tomography in high dimensions
L. Pereira, Leonardo Zambrano, J. Cortés-Vega, Sebastián Niklitschek, A. Delgado · Physical Review A · 2018
Standard quantum tomography of a single qudit achieves an infidelity that scales in the worst case as $O(1/\sqrt{N})$ for a sample of size $N$. Here, we propose a suitable generalization of the two-stage adaptive quantum tomography for a qubit to the case of a single qudit. This achieves an infidelity of the order of $O[1/\sqrt{{N}_{0}(N\ensuremath{-}{N}_{0})}]$ for all quantum states, where ${N}_{0}$ and $N\ensuremath{-}{N}_{0}$ are the ensemble sizes employed in the two stages of the method. This result is based on a second-order Taylor series expansion of the infidelity that is obtained by means of the Fr\'echet derivative and measurement outcomes modeled by a multinomial distribution. Numerical simulations indicate that the choice ${N}_{0}=N/2$ leads to an infidelity that scales approximately as $O(1/N)$ for all quantum states in a wide range of dimensions, that is, a quadratic improvement of the infidelity when compared to standard quantum tomography in the case of low-rank states.