Abstract
Accurately inferring the state of a quantum device from the results of measurements is a crucial task in building quantum information processing hardware. The predominant state estimation procedure, maximum likelihood estimation (MLE), generally reports an estimate with zero eigenvalues. These cannot be justified. Furthermore, the MLE estimate is incompatible with error bars, so conclusions drawn from it are suspect. I propose an alternative procedure, Bayesian mean estimation (BME). BME never yields zero eigenvalues, its eigenvalues provide a bound on their own uncertainties, and under certain circumstances it is provably the most accurate procedure possible. I show how to implement BME numerically, and how to obtain natural error bars that are compatible with the estimate. Finally, I briefly discuss the differences between Bayesian and frequentist estimation techniques.
Highlights
Inferring the state of a quantum device from the results of measurements is a crucial task in building quantum information processing hardware
The maximum likelihood estimation (MLE) estimate is at best sub-optimal, and at worst dangerously unreliable
MLE is a sort of minimal fix for tomography, returning the non-negative state that is in some sense ‘closest’ to ρtomo
Summary
It is impossible to bracket a zero probability with consistent error bars. The MLE estimate is at best sub-optimal, and at worst dangerously unreliable (implying, for instance, that certain errors can be ruled out). Bayesian mean estimation (BME) is an alternative procedure that avoids these pitfalls. The simple underlying principle is that the best estimate is an average over all states ρ consistent with the data, weighted by their likelihood. The BME estimate is always full-rank, and comes equipped with a natural set of error bars. BME is provably the most accurate scheme possible [5], under certain reasonable assumptions.
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