Experimental Study of Optimal Measurements for Quantum State Tomography

Quantum tomography is a critically important tool to evaluate quantum hardware, making it essential to develop optimized measurement strategies that are both accurate and efficient. We compare a variety of strategies using nearly pure test states. Those that are informationally complete for all states are found to be accurate and reliable even in the presence of errors in the measurements themselves, while those designed to be complete only for pure states are far more efficient but highly sensitive to such errors. Our results highlight the unavoidable trade-offs inherent in quantum tomography.

Figure 1

Stern-Gerlach analysis of ${}^{133}\mathrm{Cs}$ atoms in a hyperfine state with support on all 16 magnetic sublevels. (a) Time-of-flight signal from atoms in the $F=4$ manifold. (b) Same from atoms in the $F=3$ manifold. The black lines are measured signals; the light red lines are fits.

Figure 2

Average QST infidelities $\mathrm{\Delta }$ for nine POVMs in three IC categories (data from Table 1). (a) QST in $d=4$. (b) QST in $d=16$. POVMs consisting of multiple orthogonal measurements are shown in (dark) blue; POVMs consisting of a single nonorthogonal measurement with $>d$ outcomes are shown in (light) red.

Figure 3

QST of nearly pure test states in $d=16$, using data from POVMs that are less than fully IC. (a) Infidelity versus the number $N$ of mutually unbiased based used (fully IC for $N=17$). (b) Infidelity versus number of Gell-Mann bases used (fully IC for $N=31$). Both POVMs become R1S-IC at $N=5$