Adaptive quantum tomography in high dimensions

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 $O[1/\sqrt{N_0(N-N_0)}]$ for all quantum states, where $N_0$ and $N-N_0$ 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 Fréchet 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.

Figure 1

Mean infidelity $\overline{I}$ as a function of ensemble size $N$ for two-stage adaptive tomography with $N_0=N/2$ (blue squares), $N_0=N^{2/3}$ (green triangles), and standard quantum tomography (orange circles), for several dimensions $d$. Average considers uniformly distributed pure states. For $d=2,3$ and $d=4$ average infidelity was obtained from a sample of $5\times {10}^3$ states. For $d=6,8$ and $d=10$ average infidelity was obtained from a sample of ${10}^3$ states.

Figure 2

Mean infidelity $\overline{I}$ as a function of ensemble size $N$ for two-stage adaptive tomography with $N_0=N/2$ (blue squares), $N_0=N^{2/3}$ (green triangles), and standard quantum tomography (orange circles), for several dimensions $d$. Average considers uniformly distributed states of rank $r=[d/2]$. For $d=2,3$ and $d=4$ average infidelity was obtained from a sample of $5\times {10}^3$ states. For $d=6,8$ and $d=10$ average infidelity was obtained from a sample of ${10}^3$ states.

Figure 3

Mean infidelity $\overline{I}$ as a function of ensemble size $N$ for two-stage adaptive tomography with $N_0=N/2$ (blue squares), $N_0=N^{2/3}$ (green triangles), and standard quantum tomography (orange circles), for several dimensions $d$. Average considers uniformly distributed full-rank states. For $d=2,3$ and $d=4$ average infidelity was obtained from a sample of $5\times {10}^3$ states. For $d=6,8$ and $d=10$ average infidelity was obtained from a sample of ${10}^3$ states.

Figure 4

Mean infidelity $\overline{I}$ as a function of ensemble size $N$ for two-stage adaptive tomography with $N_0=N/2$ (blue squares), $N_0=N^{2/3}$ (green triangles), and standard quantum tomography (orange circles), for several dimensions $d$. Average considers pure states uniformly distributed affected by white noise Eq. (43) with $\lambda =0.99$. For $d=2,3$ and $d=4$ average infidelity was obtained from a sample of $5\times {10}^3$ states. For $d=6,8$ and $d=10$ average infidelity was obtained from a sample of ${10}^3$ states.

Figure 5

Median infidelity $\overline{I}$ as a function of ensemble size $N$ for two-stage adaptive tomography with $N_0=N/2$ (blue squares), $N_0=N^{2/3}$ (green triangles), and standard quantum tomography (orange circles), for several dimensions $d$. The median considers uniformly distributed pure states. For $d=2,3$ and $d=4$ median infidelity was obtained from a sample of $5\times {10}^3$ states. For $d=6,8$ and $d=10$ median infidelity was obtained from a sample of ${10}^3$ states.

Figure 6

Median infidelity $\overline{I}$ as a function of ensemble size $N$ for two-stage adaptive tomography with $N_0=N/2$ (blue squares), $N_0=N^{2/3}$ (green triangles), and standard quantum tomography (orange circles), for several dimensions $d$. The median considers uniformly distributed states of rank $r=[d/2]$. For $d=2,3$ and $d=4$ median infidelity was obtained from a sample of $5\times {10}^3$ states. For $d=6,8$ and $d=10$ median infidelity was obtained from a sample of ${10}^3$ states.

Figure 7

Median infidelity $\overline{I}$ as a function of ensemble size $N$ for two-stage adaptive tomography with $N_0=N/2$ (blue squares), $N_0=N^{2/3}$ (green triangles), and standard quantum tomography (orange circles), for several dimensions $d$. The median considers uniformly distributed full-rank states. For $d=2,3$ and $d=4$ median infidelity was obtained from a sample of $5\times {10}^3$ states. For $d=6,8$ and $d=10$ median infidelity was obtained from a sample of ${10}^3$ states.

Figure 8

Median infidelity $\overline{I}$ as a function of ensemble size $N$ for two-stage adaptive tomography with $N_0=N/2$ (blue squares), $N_0=N^{2/3}$ (green triangles), and standard quantum tomography (orange circles), for several dimensions $d$. The median considers pure states uniformly distributed affected by white noise Eq. (43) with $\lambda =0.99$. For $d=2,3$ and $d=4$ median infidelity was obtained from a sample of $5\times {10}^3$ states. For $d=6,8$ and $d=10$ median infidelity was obtained from a sample of ${10}^3$ states.