Error probability analysis in quantum tomography: a tool for evaluating experiments

TL;DR

This paper extends the concept of error probability to quantum tomography, deriving bounds on its decay rate, and shows how combining error probability with risk functions enhances experimental evaluation.

Contribution

It introduces a framework linking error probability with quantum tomography, deriving bounds and demonstrating how to evaluate experiments more comprehensively.

Findings

Error probability decreases at most exponentially with trials

Maximum likelihood estimator achieves the exponential bound

Identifiability aligns with informational completeness in tomography

Abstract

We expand the scope of the statistical notion of error probability, i.e., how often large deviations are observed in an experiment, in order to make it directly applicable to quantum tomography. We verify that the error probability can decrease at most exponentially in the number of trials, derive the explicit rate that bounds this decrease, and show that a maximum likelihood estimator achieves this bound. We also show that the statistical notion of identifiability coincides with the tomographic notion of informational completeness. Our result implies that two quantum tomographic apparatuses that have the same risk function, (e.g. variance), can have different error probability, and we give an example in one qubit state tomography. Thus by combining these two approaches we can evaluate, in a reconstruction independent way, the performance of such experiments more discerningly.