Minimax Quantum Tomography: Estimators and Relative Entropy Bounds

A minimax estimator has the minimum possible error (“risk”) in the worst case. We construct the first minimax estimators for quantum state tomography with relative entropy risk. The minimax risk of nonadaptive tomography scales as $O(1/\sqrt{N})$—in contrast to that of classical probability estimation, which is $O(1/N)$—where $N$ is the number of copies of the quantum state used. We trace this deficiency to sampling mismatch: future observations that determine risk may come from a different sample space than the past data that determine the estimate. This makes minimax estimators very biased, and we propose a computationally tractable alternative with similar behavior in the worst case, but superior accuracy on most states.

Figure 1

Estimators for Pauli measurements on a rebit, depicted as distortions of the linear inversion grid [see the text after Eq. (7)]. (a) Three standard estimators for $M=8$ measurements of $X$ and $Y$. Vertices of the red grid correspond to estimated states. Linear inversion estimates extend outside the Bloch disk of physical states. MLE’s estimates are non-negative; HML’s are strictly positive. (b) Minimax estimators for $M=8$, 16, 32, 64 measurements of $X$ and $Y$ on a rebit. Ripples indicate local bias toward support points of the least favorable prior [9].

Figure 2

Numerical minimax risk for qubits, rebits, and noisy coins. Black curves show the risk of numerically constructed minimax estimators for a qubit and a rebit, as a function of the number of samples ($N$), up to the maximum that was numerically feasible. Red curves illustrate the numerically computed risk of noisy coin systems whose noise levels are chosen to match the effective noise of the qubit and the rebit, respectively. Blue lines show the lower bound given in Eq. (6).

Figure 3

Maximum and pointwise risk of minimax and HML estimators. (a) The maximum risk, for qubit tomography, of the minimax estimator and three different HML estimators ($\beta =0.01$, 0.04, 0.10) for $N\le 192$ samples, distributed equally among the three Pauli bases. (b) The pointwise risk, along the axis oriented at 45° to both $X$ and $Y$, of the same estimators for $N=128$ samples for a rebit [this minimax estimator is depicted in Fig. 1]. The two local maxima of $\overline{d}(\rho )$ are at $r=1$ and $r\approx 1-1/\sqrt{N}$. Choosing $\beta \approx 0.04$ balances these risks and is therefore minimax among HML estimators. This HML estimator comes very close to matching the worst-case performance of the minimax estimator and outperforms it dramatically in the interior of the state space.