User-friendly confidence regions for quantum state tomography

Quantum state tomography is the standard technique for reconstructing a quantum state from experimental data. Given finite statistics, experimental data cannot give perfect information about the quantum state. A common way to express this limited knowledge is by providing confidence regions in state space. Though plenty of confidence regions have been previously proposed, they are often too loose to use for large systems or difficult to apply to nonstandard measurement schemes. Starting from a vector Bernstein inequality, we consider concentration bounds for random vectors following multinomial distributions and analyse optimal strategies to distribute a fixed budget of samples across them. Interpreting this as a tomography experiment leads to two confidence regions, one of which performs comparably well to the best regions in the literature. The regions describe an ellipsoid in the state space and have the appeal of being efficient in the required number of samples as well as being easily applicable to any measurement scheme.