Measuring $\mathrm{Tr}{\rho }^n$ on Single Copies of $\rho$ Using Random Measurements

While it is known that $\mathrm{Tr}{\rho }^n$ can be measured directly (i.e., without first reconstructing the density matrix) by performing joint measurements on $n$ copies of the same state $\rho$, it is shown here that random measurements on single copies suffice, too. Averaging over the random measurements directly yields estimates of $\mathrm{Tr}{\rho }^n$, even when it is not known what measurements were actually performed (so that $\rho$ cannot be reconstructed).

Figure 1

This plot shows, for a pure two-qubit state, the estimated values of $p_2$, $p_3$, and $p_4$ (blue, $p_2$; red, $p_3$; green, $p_4$) for 100 trials, each trial using just one value of $k$, containing an average over $N_{\mathrm{rand}}=100$ random unitaries, and using $N=4$ in (8). The mean standard deviations (over 100 trials) were $\Delta {\overline{p}}_2=0.282$ [note that this agrees with the result (9), since $\sqrt{(52/7)/99}\approx 0.274$], $\Delta {\overline{p}}_3=0.21$, $\Delta {\overline{p}}_2=0.29$. The mean estimates obtained by pooling all data from the 100 trials for ${\overline{p}}_n$ are ${\overline{p}}_2=0.990$, ${\overline{p}}_3=1.01$, and ${\overline{p}}_4=1.02$, which are all consistent with their mean standard deviations (10 times smaller than the $\Delta {\overline{p}}_n$ given above).

Figure 2

The same as the previous figure but using the estimate $N\approx 1/P_1(k)$ in (8). Here we have $\Delta {\tilde{p}}_2=0.09$, $\Delta {\tilde{p}}_3=0.17$, and $\Delta {\tilde{p}}_4=0.27$. The mean estimates obtained from pooling all data (which are the same “raw” data as in Fig. 1) from the 100 trials for ${\tilde{p}}_n$ are ${\tilde{p}}_2=1.005$ [which is indeed better than ${\overline{p}}_2$], ${\tilde{p}}_3=1.01$, and ${\tilde{p}}_4=1.02$, all consistent with the statistical errors in the mean (which are 10 times smaller than $\Delta {\tilde{p}}_n$).

Figure 3

Scatter plot of estimated values of $p_2$ (blue crosses), $p_3$ (red circles), and $p_4$ (green diamonds) versus their actual values for 200 randomly picked two-qubit input states. Here an average is taken over $N_{\mathrm{rand}}=30$ random unitaries, as well as over 4 measurement outcomes. For convenience, the dashed line gives the diagonal on which estimated and actual values agree.

Figure 4

The same as Fig. 3 but for randomly picked five-qubit states ($M=N=32$). Averaging over the same number of random unitaries (here $N_{\mathrm{rand}}=30$) produces a smaller statistical error for larger systems.

Figure 5

The standard deviation in the mean estimates for ${\tilde{p}}_2$ (blue crosses), ${\tilde{p}}_3$ (red circles), and ${\tilde{p}}_4$ (green diamonds) after averaging over $N_{\mathrm{rand}}=30$ random unitaries, for pure multiqubit states, as a function of the number of qubits.