Efficient Bayesian credible-region certification for quantum-state tomography

Abstract

Standard Bayesian credible-region theory for constructing an error region on the unique estimator of an unknown state in general quantum-state tomography to calculate its size and credibility relies on heavy Monte Carlo sampling of the state space followed by filtering to obtain the correct region sample. This conventional methodology typically gives negligible yield for very small error regions originating from large data sets. In this article, we discuss at length the in-region sampling theory for computing both size and credibility from region-average quantities that avoids this general problem altogether. Among the many possible numerical choices, we study the performance and properties of accelerated hit-and-run Monte Carlo algorithm for in-region sampling and provide its complexity estimates for quantum states. Finally with our in-region concept, by alternatively quantifying the region capacity with the region-average distance between two states in the region (measured for instance with either the Hilbert-Schmidt, trace-class, or Bures distance), we derive approximation formulas to analytically estimate both region capacity and credibility without Monte Carlo computation.