Adaptive Bayesian quantum tomography

In this paper we revisit the problem of optimal design of quantum tomographic experiments. In contrast to previous approaches where an optimal set of measurements is decided in advance of the experiment, we allow for measurements to be adaptively and efficiently reoptimized depending on data collected so far. We develop an adaptive statistical framework based on Bayesian inference and Shannon's information, and demonstrate a significant reduction in the total number of measurements required as compared to nonadaptive methods, including mutually unbiased bases.

Figure 1

Adaptive selection of measurements based of partial data. Scatter plots show 400 samples from current posterior. Shaded circles around the “Bloch disk” show relative value of the objective in Eq. (6) for different measurement directions (lighter is higher). Pairs of arrows show the most informative next measurement. Circular histograms show the number of times measurement directions have been used. (a) Initially, no observations are made; samples shown are from the uniform prior. All measurements are equally informative; we chose to start with $\left\{\right|H\rangle ,|V\rangle \}$. (b) After one measurement, the posterior is updated; the next best measurement is mutually unbiased with respect to the first one. It is now $\left\{\right|D\rangle ,|A\rangle \}$. (c) After two observations, the next best measurement is equally biased to the first two bases. (d) Posterior after 1000 observations concentrates around true state. The method tries a range of measurements, with a tendency to point toward the solution.

Figure 2

One qubit tomography using projective measurements. (a) Improvement of mean posterior fidelity as the experiment progresses. Results are shown for uniformly sampled measurements (), uniformly sampled Pauli measurements (), ABQT selecting adaptively among the three Pauli measurements (), and ABQT picking general measurements (). Adaptive optimization of measurements allows for an almost $n^{-1}$ rate of convergence, while other methods are more consistent with a $n^{-\frac{1}{2}}$ rate. (b) Final value of the mean posterior infidelity after 6000 measurements using the four methods as before, shown as a function of purity of the state to be estimated. The advantage of ABQT is greatest for states with higher purity.

Figure 3

Two qubit QST with uniformly chosen amongst MUB () or SSQT bases () and ABQT picking from the same set of MUBs (), SSQT bases () or a more flexible set of 81 separable bases (). Cases (a)–(c) are the same as those in [2]; (d) shows average results over 20 randomly generated entangled pure states. (a) As expected, for the maximally mixed state the choice of measurement strategy has little effect. (b) On the entangled state $\left(\right|HH\rangle +|VV\rangle )/\sqrt{2}$ MUB outperforms SSQT when uniformly sampled, but by allowing for adaptivity we can close the performance gap. (c) SSQT outperforms MUBs on the separable state $|HV\rangle$, but again, picking measurements adaptively the two sets perform similarly. (d) For random pure states a large improvement in performance is made when performing ABQT with the flexible set of separable measurements. Using this set, ABQT only needs ${10}^4$ measurements to achieve $\approx 98.7\%$ mean fidelity for which MUB needs ${10}^5$.