Error estimation in the direct state tomography

TL;DR

This paper provides an analytical framework for estimating the accuracy of Direct State Tomography (DST) using non-orthogonal basis projections, enabling error analysis across different measurement strengths including weak measurements.

Contribution

It introduces a reformulation of DST in terms of non-orthogonal basis projections, allowing explicit error estimation and analysis similar to standard quantum state reconstruction methods.

Findings

Derived an explicit formula for average minimum square errors in DST

Extended error analysis to weak measurement regimes

Enabled accuracy assessment for any measurement strength

Abstract

We show that reformulating the Direct State Tomography (DST) protocol in terms of projections into a set of non-orthogonal bases one can perform an accuracy analysis of DST in a similar way as in the standard projection-based reconstruction schemes. i.e. in terms of the Hilbert-Schmidt distance between estimated and true states. This allows us to determine the estimation error for any measurement strength, including the weak measurement case, and to obtain an explicit analytic form for the average minimum square errors.