# Two-stage Estimation for Quantum Detector Tomography: Error Analysis, Numerical and Experimental Results

## TL;DR

This paper introduces a two-stage quantum detector tomography method that improves calibration accuracy by combining linear regression with physical constraints, validated through simulations and experiments.

## Contribution

A novel two-stage estimation approach for quantum detector tomography that enhances accuracy and guarantees physical validity, with detailed error analysis and optimization.

## Key findings

- Error upper bound established for the method
- Simulation results demonstrate improved accuracy
- Experimental validation confirms effectiveness

## Abstract

Quantum detector tomography is a fundamental technique for calibrating quantum devices and performing quantum engineering tasks. In this paper, a novel quantum detector tomography method is proposed. First, a series of different probe states are used to generate measurement data. Then, using constrained linear regression estimation, a stage-1 estimation of the detector is obtained. Finally, the positive semidefinite requirement is added to guarantee a physical stage-2 estimation. This Two-stage Estimation (TSE) method has computational complexity $O(nd^2M)$, where $n$ is the number of $d$-dimensional detector matrices and $M$ is the number of different probe states. An error upper bound is established, and optimization on the coherent probe states is investigated. We perform simulation and a quantum optical experiment to testify the effectiveness of the TSE method.