# Efficient tomography with unknown detectors

## TL;DR

This paper compares standard detector tomography and data-pattern tomography, revealing regimes where each outperforms the other based on the mathematical properties of pseudoinverses, supported by numerical simulations.

## Contribution

It provides a theoretical explanation for the differences between the two techniques and identifies conditions favoring each method in quantum state tomography.

## Key findings

- Data-pattern tomography outperforms standard tomography in certain regimes.
- The difference is linked to the nonexistence of the reverse-order law for pseudoinverses.
- Numerical simulations confirm the theoretical predictions.

## Abstract

We compare the two main techniques used for estimating the state of a physical system from unknown measurements: standard detector tomography and data-pattern tomography. Adopting linear inversion as a fair benchmark, we show that the difference between these two protocols can be traced back to the nonexistence of the reverse-order law for pseudoinverses. We capitalize on this fact to identify regimes where the data-pattern approach outperforms the standard one and vice versa. We corroborate these conclusions with numerical simulations of relevant examples of quantum state tomography.