Simultaneous estimations of quantum state and detector through multiple quantum processes

TL;DR

This paper presents a unified framework using multiple quantum processes to simultaneously estimate quantum states and detectors, providing a closed-form algorithm with optimal error scaling validated by numerical tests.

Contribution

Introduces a novel framework for joint quantum state and detector estimation using multiple processes, with a closed-form solution and polynomial optimization approach.

Findings

Mean squared error scales as O(1/N) for both tasks

Framework is validated through numerical examples

Simultaneous estimation improves efficiency over separate methods

Abstract

The estimation of all the parameters in an unknown quantum state or measurement device, commonly known as quantum state tomography (QST) and quantum detector tomography (QDT), is crucial for comprehensively characterizing and controlling quantum systems. In this paper, we introduce a framework, in two different bases, that utilizes multiple quantum processes to simultaneously identify a quantum state and a detector. We develop a closed-form algorithm for this purpose and prove that the mean squared error (MSE) scales as $O(1/N) $ for both QST and QDT, where $N $ denotes the total number of state copies. This scaling aligns with established patterns observed in previous works that addressed QST and QDT as independent tasks. Furthermore, we formulate the problem as a sum of squares (SOS) optimization problem with semialgebraic constraints, where the physical constraints of the state and detector are characterized by polynomial equalities and inequalities. The effectiveness of our proposed methods is validated through numerical examples.