Precision-Guaranteed Quantum Tomography

Quantum state tomography is currently the standard tool for verifying that a state prepared in the lab is close to an ideal target state, but up to now there have been no rigorous methods for evaluating the precision of the state preparation in tomographic experiments. We propose a new estimator for quantum state tomography, and prove that the (always physical) estimates will be close to the true prepared state with a high probability. We derive an explicit formula for evaluating how high the probability is for an arbitrary finite-dimensional system and explicitly give the one- and two-qubit cases as examples. This formula applies for any informationally complete sets of measurements, arbitrary finite number of data sets, and general loss functions including the infidelity, the Hilbert-Schmidt, and the trace distances. Using the formula, we can evaluate not only the difference between the estimated and prepared states, but also the difference between the prepared and target states. This is the first result directly applicable to the problem of evaluating the precision of estimation and preparation in quantum tomographic experiments.

Figure 1

Three-way relationship: ${\rho }_{*}$ is a target state we want to prepare, $\rho$ is the true prepared state, and ${\rho }_N^{\mathrm{est}}$ is an estimate made from $N$ tomographic data sets.

Figure 2

Confidence level of ${\rho }^{\mathrm{ENM}}$ for the error threshold $\delta =0.07$ in quantum state tomography: (a) is the one-qubit ($k=1$) case and (b) is for two qubits ($k=2$). The line styles are fixed as follows: solid (black) line for detection efficiency $\eta =1$, dashed (red) line for $\eta =0.9$, dot-dashed (blue) line for $\eta =0.8$.